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ENGINEERING   MATHEMATICS 


ENGmEERING  MATHEMATICS 


A  SERIES  OF  LECTURES  DELIVERED 
AT  UNION  COLLEGE 


BY 


CHARLES  PROTEUS  STEINMETZ,  A.M.,  Ph.D„ 

PAST    PRESIDENT 
AMERICAN   INSTITUTE    OF    ELECTRICAL    ENGINEERS 


Third  Edition 

Revised  and  EisLARciED 

Eighth  Imi'ressiun 


McGRAW-HILL  BOOK  COMPANY,  Inc. 

NEW   YORK:    370   SEVENTH   AVENUE 

LONDON:    6*8  BOUVERIE  ST.,  E.  C.  4 

1917 


Copyright,  1911,  1915  and  1917,  by  the 
McGraw-Hill  Book  Company,  Inc. 


PRINTED  IN   THE    UNITED   STATES    OF   AMERICA 


PREFACE  TO  THIRD  EDITION. 


In  preparing  the  third  edition  of  Engineering  Mathematics, 
besides  revision  and  correction  of  the  previous  text,  considera- 
ble new  matter  has  been  added. 

The  chain  fraction  has  been  recognized  and  discussed  as  a 
convenient  method  of  numerical  representation  and  approxi- 
mation; a  paragraph  has  been  devoted  to  the  diophantic  equa- 
tions, and  a  section  added  on  engineering  reports,  discussing 
the  different  purposes  for  which  engineering  reports  are  made, 
and  the  corresponding  character  and  nature  of  the  report,  in 
its  bearing  on  the  success  and  recognition  of  the  engineer's 
work. 

Charles  Proteus  Steinmbtz. 
Camp  Mohawk, 

September  1st,  1917. 


PREFACE  TO  FIRST  EDITION. 


The  following  work  embodies  the  subject-matter  of  a  lecture 
course  which  I  have  given  to  the  junior  and  senior  electrical 
engineering  students  of  Union  University  for  a  number  of 
years. 

It  is  generally  conceded  that  a  fair  knowledge  of  mathe- 
matics is  necessary  to  the  engineer,  and  especially  the  electrical 
engineer.  For  the  latter,  however,  some  branches  of  mathe- 
matics are  of  fundamental  importance,  as  the  algebra  of  the 
general  number,  the  exponential  and  trigonometric  series,  etc., 
which  are  seldom  adequately  treated,  and  often  not  taught  at 
all  in  the  usual  text-books  of  mathematics,  or  in  the  college 
course  of  analytic  geometry  and  calculus  given  to  the  engineer- 
ing students,  and,  therefore,  electrical  engineers  often  possess 
little  knowledge  of  these  subjects.  As  the  result,  an  electrical 
engineer,  even  if  he  possess  a  fair  knowledge  of  mathematics, 
may  often  find  difficulty  in  dealing  with  problems,  through  lack 
of  familiarity  with  these  branches  of  mathematics,  which  have 
become  of  importance  in  electrical  engineering,  and  may  also 
find  difficulty  in  looking  up  information  on  these  subjects. 

In  the  same  way  the  college  student,  when  beginning  the 
study  of  electrical  engineering  theory,  after  completing  his 
general  course  of  mathematics,  frequently  finds  himself  sadly 
deficient  in  the  knowledge  of  mathematical  subjects,  of  which 
a  complete  familiarity  is  required  for  effective  understanding 
of  electrical  engineering  theory.  It  was  this  experience  which 
led  me  some  years  ago  to  start  the  course  of  lectures  which 
is  reproduced  in  the  following  pages.  I  have  thus  attempted  to 
bring  together  and  discuss  expHcitly,  with  numerous  practical 
applications,  all  those  branches  of  mathematics  which  are  of 
special  importance  to  the  electrical  engineer.     Added  thereto 

vii 


viii  PREFACE. 

are  a  number  of  subjects  ^hich  experience  has  shown  me 
to  be  important  for  the  effective  and  expeditious  execution  of 
electrical  engineering  calculations.  Mere  theoretical  knowledge 
of  mathematics  is  not  sufficient  for  the  engineer,  but  it  must 
be  accompanied  by  ability  to  apply  it  and  derive  results — to 
carry  out  numerical  calculations.  It  is  not  sufficient  to  know 
how  a  phenomenon  occurs,  and  how  it  may  be  calculated,  but 
very  often  there  is  a  wide  gap  between  this  knowledge  and  the 
ability  to  carry  out  the  calculation;  indeed,  frequently  an 
attempt  to  apply  the  theoretical  knowledge  to  derive  numerical 
results  leads,  even  in  simple  problems,  to  apparently  hopeless 
complication  and  almost  endless  calculation,  so  that  all  hope 
of  getting  rehable  results  vanishes.  Thus  considerable  space 
has  been  devoted  to  the  discussion  of  methods  of  calculation, 
the  use  of  curves  and  their  evaluation,  and  other  kindred 
subjects  requisite  for  effective  engineering  work. 

Thus  the  following  work  is  not  intended  as  a  complete 
course  in  mathematics,  but  as  supplementary  to  the  general 
college  course,  of  mathematics,  or  to  the  general  knowledge  of 
mathematics  which  eveiy  engineer  and  really  every  educated 
man  should  possess. 

In  illustrating  the  mathematical  discussion,  practical 
examples,  usually  taken  from  the  field  of  electrical  engineer- 
ing, have  been  given  and  discussed.  These  are  sufficiently 
numerous  that  any  example  dealing  with  a  phenomenon 
with  which  the  reader  is  not  yet  familiar  may  be  omitted  and 
taken  up  at  a  later  time. 

As  appendix  is  given  a  descriptive  outline  of  the  intro- 
duction to  the  theory  of  functions,  since  the  electrical  engineer 
should  be  familiar  with  the  general  relations  between  the 
different  functions  which  he  meets. 

In  relation  to  "  Theoretical  Elements  of  Electrical  Engineer- 
ing," "  Theory  and  Calculation  of  Alternating  Current  Phe- 
nomena," and  "  Theoiy  and  Calculation  of  Transient  Electric 
Phenomena,"  the  following  work  is  intended  as  an  introduction 
and  explanation  of  the  mathematical  side,  and  the  most  efficient 
method  of  study,  appears  to  me,  to  start  with  "  Electrical 
Engineering  Mathematics,"  and  after  entering  its  third 
chapter,  to  take  up  the  reading  of  the  first  section  of  "  Theo- 
retical Elements,"  and  then  parallel  the  study  of  "  Electrical 


PREFACE.  IX 

Engineering  Mathematics,"  "  Theoretical  Elements  of  Electrical 
Engineering,"  and  "  Theory  and  Calculation  of  Ahernating 
Current  Phenomena,"  together  with  selected  chapters  from 
"  Theory  and  Calculation  of  Transient  Electric  Phenomena," 
and  after  this,  once  more  systematically  go  through  all  four 

books. 

Charles  P.  Steinmetz. 

Schenectady,  N.  Y., 
December,  1910. 


PREFACE  TO  SECOND  EDITION. 


In  preparing  the  second  edition  of  Engineering  Mathe- 
matics, besides  revision  and  correction  of  the  previous  text, 
considerable  new  matter  has  been  added,  more  particularly 
with  regard  to  periodic  curves.  In  the  former  edition  the 
study  of  the  wave  shapes  produced  by  various  harmonics, 
and  the  recognition  of  the  harmonics  from  the  w^ave  shape, 
have  not  been  treated,  since  a  short  discussion  of  wave  shapes 
is  given  in  "Alternating  Current  Phenomena."  Since,  how- 
ever, the  periodic  functions  are  the  most  important  in  elec- 
trical engineering,  it  appears  necessary  to  consider  their  shape 
more  extensively,  and  this  has  been  done  in  the  new  edition. 

The  symbolism  of  the  general  number,  as  applied  to  alter- 
nating waves,  has  been  changed  in  conformity  to  the  decision 
of  the  International  Electrical  Congress  of  Turin,  a  discussion 
of  the  logarithmic  and  semi-logarithmic  scale  of  curve  plot- 
ting given,  etc. 

Charles  P.  Steinmetz. 
December,  1914. 


CONTENTS. 


FAQB 

Preface v 

CHAPTER   I.     THE   GENERAL   NUMBER. 

A.  The  System  of  Numbers. 

1.  Addition  and  Subtraction.     Origin  of  numbers.     Counting  and 

measuring.     Addition.     Subtraction  as   reverse  operation  of 
addition 1 

2.  Limitation  of  subtraction.    Subdivision  of  the  absolute  numbers 

into  positive  and  negative 2 

3.  Negative  number  a  mathematical  conception  like  the  imaginary 

number.     Cases  where  the  negative  number  has  a  physical 
meaning,  and  cases  where  it  has  not 4 

4.  Multiplication  and  Division.     Multiplication  as  multiple  addi- 

tion.    Division  as  its  reverse  operation.     Limitation  of  divi- 
sion         6 

5.  The  fraction  as  mathematical  conception.     Cases  where  it  has  a 

physical  meaning,  and  cases  where  it  has  not 8 

6.  Involution  and  Evolution.     Involution  as  multiple  multiplica- 

tion.    Evolution  as  its  reverse  operation.      Negative    expo- 
nents        9 

7.  Multiple  involution  leads  to  no  new  operation 10 

8.  Fractional  exponents 10 

9.  Irrational  Numbers.    Limitation  of  evolution.    Endless  decimal 

fraction.     Rationality  of  the  irrational  number 11 

1 0.  Quadrature  numbers.     Multiple  values  of  roots.     Square  root  of 

negative  quantity  representing  quadrature  number,  or  rota- 
tion by  90° 13 

11.  Comparison   of   positive,    negative    and    quadrature   numbers. 

Reality  of  quadrature  number.     Cases  where  it  has  a  physical 
meaning,  and  cases  where  it  has  not 14 

12.  General  Numbers.     Representation  of  the  plane  by  the  general 

number.     Its  relation  to  rectangular  coordinates 16 

13.  Limitation  of  algebra  by  the  general  number.     Roots  of  the  unit. 

Number  of  such  roots,  and  their  relation 18 

14.  The  two  reverse  operations  of  involution 19 

xi 


xii  CONTENTS. 

PAGE 

15.  Logarithmation.     Relation  between  logarithm  and  exponent  of 

involution.  Reduction  to  other  base.  Logarithm  of  negative 
quantity 20 

16.  Quaternions.     Vector  calculus  of  space 22 

17.  Space  rotors  and  their  relation.     Super  algebraic  nature  of  space 

analysis 22 

B.  Algp:bra  of  the  General  Number  of  Complex  Quantity. 

Rectangular  and  Polar  Coordinates 25 

18.  Powers  of  /.     Ordinary  or  real,  and  quadrature  or  imaginary 

number.     Relations 25 

19.  Conception  of  general. number  by  point  of  plane  in  rectangular 

coordinates;  in  polar  coordinates.  Relation  between  rect- 
angular and  polar  form 26 

20.  Addition  and  Subtraction.     Algebraic  and  georretrical  addition 

and  subtraction.  Combination  and  resolution  by  parallelo- 
gram law 28 

21.  Denotations 30 

22.  Sign  of  vector  angle.     Conjugate  and  associate  numbers.     Vec- 

tor analysis 30 

23.  Instance  of  steam  path  of  turbine 33 

24.  Multiplication.     Multiplication  in  rectangular  coordinates.  ...  38 

25.  Multiplication  in  polar  coordinates.     Vector  and  operator 38 

26.  Physical  meaning  of  result  of  algebraic  operation.     Representa- 

tion of  result 40 

27.  Limitation  of  application  of  algebraic  operations  to  physical 

quantities,  and  of  the  graphical  representation  of  the  result. 
Graphical  representation  of  algebraic  operations  between 
current,  voltage  and  impedance 40 

28.  Representation  of  vectors  and  of  operators 42 

29.  Division.     Division  in  rectangular  coordinates 42 

30.  Division  in  polar  coordinates 43 

31.  Livolution  and  Evolution.     Use  of  polar  coordinates 44 

32.  Multiple  values  of  the  result  of  evolution.    Their  location  in  the 

plane  of  the  general  number.     Polyphase  and  n  phase  systems 

of  numbers 45 

33.  The  n  values  of  Vl  and  their  relation 46 

34.  Evolution  in  rectangular  coordinates.     Complexity  of  result ...     47 

35.  Reduction  of  products  and  fractions  of  general  numbers  by  polar 

representation.     Instance 48 

36.  Exponential  representations  of  general  numbers.    The  different 

forms  of  the  general  number 49 

37.  Instance  of  use  of  exponential  form  in  solution  of  differential 

eauation 50 


CONTENTS.  xiii 

PAGE 

38.  Logarithmation,     Resolution   of   the   logarithm   of   a   general 

number 51 

CHAPTER  11.     THE   POTENTIAL  SERIES   AND  EXPONENTIAL 
FUNCTION. 

A.  General. 

39.  The  infinite  series  of  powers  of  a; 52 

40.  Approximation  by  series 53 

41.  Alternate  and  one-sided  approximation 54 

42.  Convergent  and  divergent  series 55 

43.  Range  of  convergency.     Several   series  of  different  ranges  for 

same  expression 56 

44  Discussion  of  convergency  in  engineering  applications 57 

45.  Use  of  series  for  approximation  of  small  terms.     Instance  of 

electric  circuit 58 

46.  Binomial  theorem  for  development  in  series.      Instance  of  in- 

ductive circuit 59 

47.  Necessity  of  development  in  series.     Instance  of  arc  of  hyperbola  60 

48.  Instance  of  numerical  calculation  of  log  (l+x) 63 

B.  Differential  Equations. 

49.  Character  of  most  differential  equations  of  electrical  engineering. 

Their  typical   forms 64 

dy 

50.  -T-  =  y-     Solution  by  series,  by  method  of  indeterminate    co- 
ax 

efficients 65 

dz 

51.  -r-  =  oz.     Solution  by  indeterminate  coefficients 68 

dx 

52.  Integration  constant  and  terminal  conditions 68 

53.  Involution  of  solution.     Exponential  function 70 

54.  Instance  of  rise  of  field  current  in  direct  current  shunt  motor  .  .  72 

55.  Evaluation  of  inductance,  and  numerical  calculation 75 

56.  Instance  of  condenser  discharge  through  resistance 76 

57.  Solution  of  —  =  ay  by  indeterminate  coefficients,  by  exponential 

function 78 

58.  Solution  by  trigonometric  functions 81 

59.  Relations  between  trigonometric  functions  and  exponential  func- 

tions with  imaginary  exponent,  and  inversely 83 

60.  Instance  of  condenser  discharge  through  inductance.    The  two 

integration  constants  and  terminal  conditions 84 

61.  Effect  of  resistance  on  the  discharge.     The  general  differential 

equation 86 


xiv  CONTENTS. 

PAGE 

62.  Solution  of  the  general  differential  equation  by  means  of  the 

exponential    function,    by    the    method    of    indeterminate 
coefficients -     86 

63.  Instance  of  condenser  discharge  through  resistance  and  induc- 

tance.    Exponential  solution  and  evaluation  of  constants.  .  ..  88 
.  64.  Imaginary  exponents  of  exponential  functions.     Reduction  to 

trigonometric  functions.     The  oscillating  functions 91 

65.  Explanation  of  tables  of  exponential   functions) 92 

CHAPTER   III.     TRIGONOMETRIC    SERIES. 

A.  Trigonometric  Functions. 

66.  Definition  of  trigonometric  functions  on  circle  and  right  triangle  94 

67.  Sign  of  functions  in  different  quadrants 95 

68.  Relations  between  sin,  cos,  tan  and  cot .....  97 

69.  Negative,  supplementary  and  complementary  angles 98 

70.  Angles  (x±;:)  and  (^±^) 100 

71.  Relations  between  two  angles,  and  between  angle  and  double 

angle 1 02 

72.  Differentiation    and    integration    of    trigonometric    functions. 

Definite  integrals 103 

73.  The  binomial  relations 104 

74.  Polyphase  relations 104 

75.  Trigonometric  formulas  of  the  triangle 105 

li.  Trigonometric  Series. 

76.  Constant,  transient  and  periodic  phenomena.     Univalent  peri- 

odic function  represented  by  trigonometric  series 106 

77.  Alternating  sine  waves  and  distorted  waves 107 

78.  Evaluation  of  the  Constants  from  Instantaneous  Values.     Cal- 

culation of  constant  term  of  series 108 

79.  Calculation  of  cos-coefficients 110 

80.  Calculation  of  sin-coefficients 113 

81.  Instance  of  calculating  11th  harmonic  of  generator  wave 114 

82.  Discussion.     Instance  of  complete  calculation  of  pulsating  cur- 

rent wave 116 

83.  Alternating  waves  as  symmetrical  waves.     Calculation  of  sym- 

metrical wave    117 

84.  Separation  of  odd  and  even  harmonics  and  of  constant  term  .  . .  120 

85.  Separation  of  sine  and  cosine  components 121 

86.  Separation  of  wave  into  constant  term  and  4  component  waves  122 

87.  Discussion  of  calculation 123 

88.  Mechanism  of  calculation 124 


CONTENTS.  XV 

PAQK 

89.  Instance  of  resolution  of  the  annual  temperature  curve 125 

90.  Constants  and  equation  of  temperature  wave 131 

91.  Discussion  of  temperature  wave 132 

C.  Reduction  of  Trigonometric  Series  by  Polyphase  Relation. 

92.  Method   of   separating  certain   classes  of  harmonics,    and   its 

limitation 134 

93.  Instance  of  separating  the  3d  and  9th  harmonic  of  transformer 

exciting  current 136 

D.  Calculation  of  Trigonometric  Series  from  other  Trigono- 

metric Series. 

94.  Instance  of  calculating  current  in  long  distance  transmission  line, 

due  to  distorted  voltage  wave  of  generator.     Line  constants.  .   139 

95.  Circuit  equations,  and  calculation  of  equation  of  current ,  141 

96.  Effective  value  of  current,  and  comparison  with  the  current 

produced  by  sine  wave  of  voltage 143 

97.  Voltage  wave  of  reactance  in  circuit  of  this  distorted  current  ...   145 

CHAPTER   IV.     MAXIMA   AND   MINIMA. 

98.  Maxima  and  minima  by  curve  plotting.     Instance  of  magnetic 

permeability.     Maximum  power  factor  of  induction  motor  as 
function  of  load 147 

99.  Interpolation  of  maximum  value  in  method  of  curve  plotting. 

Error  in  case  of  unsymmetrical  curve.     Instance  of  efficiency 

of  steam  turbine  nozzle.     Discussion 149 

100.  Mathematical    method.     Maximum,    minimum    and    inflexion 

point.     Discussion 152 

101.  Instance:     Speed   of   impulse    turbine    wheel    for    maximum 

efficiency.     Current  in  transformer  for  maximum  efficiency.  154 

102.  Effect  of  intermediate  variables.     Instance:   Maximum  power 

in  resistance  shunting  a  constant  resistance  in  a  constant  cur- 
rent circuit 155 

103.  Simplification  of  calculation  by  suppression  of  unnecessary  terms, 

etc.     Instance 157 

104.  Instance:   Maximum  non-inductive  load  on  inductive  transmis- 

sion line.     Maximum  current  in  line 158 

105.  Discussion  of  physical  meaning  of  mathematical  ext-remum. 

Instance 160 

106.  Instance :  External  reactance  giving  maximum  output  of  alter- 

nator at  constant  external  resistance  and  constant  excitation. 
Discussion 161 

107.  Maximum  efficiency  of  alternator  on  non-inductive  load.     Dis- 

cussion of  physical  limitations 162 


XVI  CONTENTS. 

PA.QP 

108.  Fxtrema  with  several  independent  variables.     Method  of  math 

ematical  calculation,  and  geometrical  meaning 163 

109.  Resistance  and  reactance  of  load  to  give  maximum  output  of 

transmission  line,  at  constant  supply  voltage 165 

110.  Discussion  of  physical  limitations 167 

111.  Determination  of  extrema  by  plotting  curve  of  differential  quo- 

tient.    Instance:   Maxima  of  current  wave  of  alternator  of 
distorted  voltage  on  transmission  line 168 

112.  Graphical  calculation  of  differential  curve  of  empirical  curve, 

for  determining  extrema 170 

113.  Instance:  Maximum  permeabilit.y  calculation 170 

114.  Grouping  of  battery  cells  for  maximum  power  in  constant  resist- 

ance     171 

115.  Voltage  of  transformer  to  give   maximum  output  at  constant 

loss 1 73 

116.  Voltage  of  transformer,  at  constant  output,  to  give  maximum 

efficiency  at  full  load,  at  half  load 174 

117.  Maximum    value   of   charging   current   of   condenser   through 

inductive  circuit  (a)  at  low  resistance;   (b)  at  high  resistance.  175 

118.  At  what  output  is  the  efficiency  of  an  induction  generator  a  max- 

imum?     177 

1 1 9.  Discussion  of  physical  limitations.     Maximum  efficiency  at  con- 

stant current  output 178 

120.  Method  of  Lrast  Squares.     Relation  of  number  of  observa- 

tions  to   number    of    constants.     Discussion    of    errors  of 
observation 179 

121.  Probability  calculus  and  the  minimum  sum  of  squares  of  the 

errors 181 

122.  The  differential  equations  of  the  sura  of  least  squares 182 

123.  Instance:    Reduction  of  curve  of  power  of  induction  motor 

running    light,    into    the    component    losses.     Discussion    of 

results 182 

123A.  Diophantic  equations 186 

CHAPTER   V.     METHODS   OF   APPROXIMATION. 

124.  Frequency  of  small  quantities  in  electrical  engineering  problems. 

Instances.     Approximation  by  dropping  terms  of  higher  order.  187 

125.  Multiplication  of  terms  with  small  quantities 188 

126.  Instance  of  calculation  of  power  of  direct  current  shunt  motor  .  189 

127.  Small  quantities  in  denominator  of  fractions 190 

128.  Instance  of  calculation  of  induction  motor  current,  as  function 

of  sliu 191 


CONTENTS. 


PAGE 


129.  Use  of  binomial  series  in  approximations  of  powers  and  roots, 

and  in  numerical  calculations 193 

130.  Instance  of  calculation  of  current  in  alternating  circuit  of  low 

inductance.     Instance  of  calculation  of  short  circuit  current 

of  alternator,  as  function  of  speed 195 

131.  Use  of  exponential  series  and  logarithmic  series  in  approxima- 

tions     196 

132.  Approximations  of  trigonometric  functions 198 

133.  McLaurin's  and  Taylor's  series  in  approximations 198 

134.  Tabulation  of  various  infinite  series  and  of  the  approximations 

derived  from  them 199 

135.  Estimation   of   accuracy   of    approximation.     Application    to 

short  circuit  current  of  alternator 200 

136.  Expressions  which  are  approximated  by  (1  +s)  and  by  (1  — s) .  .   201 

137.  Mathematical  instance  of  approximation 203 

138.  Equations  of  the  transmission  Line.     Integration  of  the 

differential  equations 204 

139.  Substitution  of  the  terminal  conditions 205 

140.  The  approximate  equations  of  the  transmission  line 206 

141.  Numerical  instance.    Discussionof  accuracy  of  approximation.  207 

141A.  Approximation  by  chain  fraction 208 

141B.  Approximation  by  chain  fraction  208c 

CHAPTER   VI.     EMPIRICAL   CURVES. 

A.  General. 

142.  Relation   between   empirical  curves,  empirical  equations  and 

rational  equations 209 

143.  Physical  nature  of  phenomenon.     Points  at  zero  and  at  infinity. 

Periodic  or  non-periodic.     Constant  terms.     Change  of  curve 
law.     Scale 210 

B.  Non-Periodic   Curves. 

144.  Potential  Series.     Instance  of  core-loss  curve 212 

145.  Rational  and  irrational  use  of  potential  series.     Instance  of  fan 

motor  torque.     Limitations  of  potential  series 214 

146.  Parabolic  and  Hyperbolic  Curves.     Various  shapes  of  para- 

bolas and  of  hyperbolas 216 

147.  The  characteristic  of  parabolic  and  hyperbolic  curves.     Its  use 

and  limitation  by  constant  terms 223 

148.  The  logarithmic  characteristic.     Its  use  and  limitation 224 

149.  Exponential   and   Logarithmic   Curves.      The  exponential 

function 227 

150.  Characteristics  of  the  exponential  curve,  their  use  and  limitation 

by  constant  term.      Comparison  of  exponential    curve    and 
hyperbola 228 


xviii  CONTENTS. 

151.  Double  exponential  functions.    Various  shapes  thereof 231 

152.  Evaluation    of    Empirical    Curves.     General   principles   of 

investigation  of  empirical  curves 233 

153.  Instance :  The  volt-ampere  characteristic  of  the  tungsten  lamp, 

reduced   to   parabola   with  exponent  0.6.     Rationalized  by 
reduction  to  radiation  law 235 

154.  The  volt-ampere  characteristic  of  the  magnetite  arc,  reduced 

to  hyperbola  with  exponent  0.5 238 

155.  Change  of  electric  current  with  change  of  circuit  conditions, 

reduced  to  double  exponential  function  of  time 241 

156.  Rational  reduction  of    core-loss  curve  of  paragraph   144,  by 

parabola  with  exponent  1.6 244 

157.  Reduction  of  magnetic  characteristic,  for  higher  densities,  to 

hyperboUc  curve.    Instance  of  the  investigation  of  a  hys- 
teresis curve  of  silicon  steel 246 

C.  Periodic  Curves. 

158.  Distortion  of  sine  wave  by  harmonics 255 

159.  Third  and   fifth    harmonic.     Peak,  multiple    peak,   flat    top  and 

sawtooth 255 

160.  Combined  effect  of  third  and  fifth  harmonic 263 

161.  Even    harmonics.     Unequal    shape    and    length    of    half  waves. 

Combined  second  and  third  harmonic 266 

162.  Effect  of  high  harmonics 269 

163.  Ripples    and    nodes   caused   by    higher   harmonics.     Incommen- 

surable waves 271 

CHAPTER  VII.    NUMERICAL  CALCULATIONS. 

164.  Method  of  Calculation.     Tabular  form  of  calculation 275 

165.  Instance  of  transmission  line  regulation 277 

166.  Exactness  op  Calculation.     D^^rees  of  exxctrisss:  magnitude, 

approximate,  exact 279 

167.  Number  of  decimals 28 1 

168.  Intelligibility    of    Engineering    Data.     Curve  plotting  for 

showing  shape  of  function,  and  for  record  of  numerical  values  283 

169.  Soale  of  curves.     Principles 286 

170.  Logarithmic  and  semi-logarithmic  paper  and  its  proper  use 287 

171.  Completeness  of  record 290 

171A.  Engineering  Reports 290 

172.  Reliability  of  Numercial  Calculations.     Necessity  of  relia- 

bility in  engineering  calculations 293 

173.  Methods  of  checking  calculations.    Curve  plotting 293a 

174.  Some  frequent  errors 293b 

APPENDIX  A.    NOTES  ON  THE  THEORY  OF  FUNCTIONS. 

A.   General  Functions. 

175.  Implicit  analytic  function.     Explicit  analytic  function.     Reverse 

function 294 


CONTENTS.  XIX 

PAGE 

176.  Rational     function.      Integer     function.       Approximations     by 

Taylor's  Theorem 295 

177.  Abelian   integrals   and  Abelian  functions.     Logarithmic    integral 

and  exponential  functions 296 

178.  Trigonometric  integrals  and  trigonometric  functions.     Hyperbolic 

integrals  and  hyperbolic  functions 297 

179.  Elliptic  integrals  and  elliptic  functions.     Their  double  periodicity  298 

180.  Theta  functions.     Hyperelliptic  integrals  and  functions 300 

181.  Elliptic  functions  in  the  motion  of  the  pendulum  and  the  surging 

of  synchronous  machines 301 

182.  Instance  of  the  arc  of  an  ellipsis 301 

B.  Special  Functions. 

183.  Infinite  summation  series.     Infinite  product  series 302 

184.  Functions  by  integration.     Instance  of  the  propagation  functions 

of  electric  waves  and  impulses 303 

185.  Functions  defined  by  definite  integrals 305 

186.  Instance  of  the  gamma  function 306 

C.  Exponential,  Trigonometric  and  Hyperbolic  Functions. 

187.  Functions  of  real  variables 306 

188.  Functions  of  imaginary  variables 308 

189.  Functions  of  complex  variables 308 

190.  Relations 309 

APPENDIX  B.    TABLES. 

Table    I.    Three  decimal  exponential  functions 312 

Table  II.    Logarithms  of  exponential  functions 

Exponential  functions 313 

Hyperbolic  functions 314 

Index 315 


ENGINEERING  MATHEMATICS. 


CHAPTER   I. 
THE  GENERAL  NUMBER. 

A.  THE  SYSTEM  OF  NUMBERS. 
Addition  and  Subtraction. 

1.  From  the  operation  of  counting  and  measuring  arose  the 
art  of  figuring,  arithmetic,  algebra,  and  finally,  more  or  less, 
the  entire  structure  of  mathematics. 

During  the  development  of  the  human  race  throughout  the 
ages,  which  is  repeated  by  every  chikl  during  the  first  years 
of  life,  the  first  conceptions  of  numerical  values  were  vague 
and  crude:  many  and  few,  big  and  Httle,  large  and  small. 
Later  the  ability  to  count,  that  is,  the  knowledge  of  numbers, 
developed,  and  last  of  all  the  ability  of  measuring,  and  even 
up  to-day,  measuring  is  to  a  considerable  extent  done  by  count- 
ing: steps,  knots,  etc. 

Fi-om  counting  arose  the  simplest  arithmetical  operation — 
addition.    Thus  we  may  count  a  bunch  of  horses: 

1,  2,  3,  4,  5, 

and  then  count  a  second  bunch  of  horses, 

1    ">    3- 

J.,  _,  .», 

now  put  the  second  bunch  together  with  the  first  one,  into  one 
bunch,  and  count  them.     That  is,  after  counting  the  horees 


2  ENGINEERING  MATHEMATICS. 

of  the  first  bunch,  we  continue  to  count  those  of  the  second 
bunch,  thus: 

1,  2,  3,  4,  5-6,  7,  8; 

which  gives  addition, 

5  +  3  =  8; 
or,  in  general, 

a  +  b  =  c. 

We  may  take  away  again  the  second  bunch  of  horses,  that 
means,  we  count  the  entire  bunch  of  horses,  and  then  count 
off  those  we  take  away  thus: 

1,  2,  3,  4,  5,  6,  7,  8-7,  6,  5; 

which  gives  subtraction, 

8-3  =  5; 
or,  in  genera], 

c—b  =  a. 

The  reverse  of  putting  a  group  of  things  together  with 
another  group  is  to  take  a  group  away,  therefore  subtraction 
is  the  reverse  of  addition. 

2.  Immediately  we  notice  an  essential  difference  between 
addition  and  subtraction,  which  may  be  illustrated  by  the 
following  examples : 

Addition :  5  horses  +  3  horses  gives  8  horses, 
Subtraction:  5  horses —3  horses  gives  2  horses, 
Addition:  5  horses +7  horses  gives  12  horses. 
Subtraction:  5  horses  —7  horses  is  impossible. 

From  the  above  it  follows  that  we  can  always  add,  but  we 
cannot  always  subtract;  subtraction  is  not  always  possible; 
it  is  not,  when  the  number  of  things  which  we  desire  to  sub- 
tract is  greater  than  the  number  of  things  from  which  we 
desire  to  subtract. 

The  same  relation  obtains  in  measuring;  we  may  measure 
a  distance  from  a  starting  point  A  (Fig.  1),  for  instance  in  steps, 
and  then  measure  a  second  distance,  and  get  the  total  distance 
from  the  starting  point  by  addition:    5  steps,  from  A  to  B. 


THE  GENERAL  NUMBER. 


then  3  steps,  from  B  to  C,  gives  the  distance  from  A  to  C,  as 
8  steps. 

5  steps +  3  steps  =  8  steps; 


-t- 


12        3        4        5        6        7 
H 1 1 1        (D        I »- 


B 
Fig.  1.     Addition. 

or,  we  may  step  off  a  distance,  and  then  step  back,  that  is, 
subtract  another  distance,  for  instance  (Fig.  2), 

5  steps— 3  steps  =  2  steps; 

that  is,  going  5  steps,  from  A  to  B,  and  then  3  steps  back, 
from  B  to  C,  brings  us  to  C,  2  steps  away  from  A. 


12        3        4        5 
H J)         I 1         (D 


C  B 

Fig.  2.     Subtraction. 

Trying  the  case  of  subtraction  which  was  impossible,  in  the 
example  with  the  horses,  5  steps— 7  steps  =  ?  We  go  from  the 
starting  point.  A,  5  steps,  to  B,  and  then  step  back  7  steps; 
here  we  find  that  sometimes  we  can  do  it,  sometimes  we  cannot 
do  it;  if  back  of  the  starting  point  A  is  a  stone  wall,  we  cannot 
step  back  7  steps.  If  A  is  a  chalk  mark  in  the  road,  we  may 
step  back  beyond  it,  and  come  to  C  in  Fig.  3.     In  the  latter  case. 


■^ 


2        1        0        1        2        3        4        5 
(t)        I d)        I 1 1 1         (D 


C  A  B 

1  iG.  ?>.     Hdiniuction,  Negative  Result. 

at  C  we  arc  again  2  steps  distant  from  the  starting  point,  just 
as  in  Fig.  2.     That  is, 

5-3  =  2     (Fig.  2), 

5-7  =  2     (Fig.  3). 

In  the  case  where  we  can  subtract  7  from  5,  we  get  the  same 
distance  from  the  starting  point  as  when  we  subtract  3  from  5, 


4  ENGINEERING  MATHEMATICS. 

but  the  distance  AC  in  Fig.  3,  while  the  same,  2  steps,  as 
in  Fig.  2,  is  different  in  character,  the  one  is  toward  the  left, 
the  other  toward  the  right.  That  means,  we  have  two  kinds 
of  distance  units,  those  to  the  right  and  those  to  the  left,  and 
have  to  find  some  way  to  distinguish  them.  The  distance  2 
in  Fig.  3  is  toward  the  left  of  the  starting  point  A,  that  is, 
in  that  direction,  in  which  we  step  when  subtracting,  and 
it  thus  appears  natural  to  distinguish  it  from  the  distance 
2  in  Fig.  2,  by  calling  the  former— 2,  while  we  call  the  distance 
AC  in  Fig.  2:  +2,  since  it  is  in  the  direction  from  A,  in  which 
we  step  in  adding. 

This  leads  to  a  subdivision  of  the  system  of  absolute  numbers, 

1,  2,  3,... 

into  two  classes,  positive  mmibers, 

+  1,    +2,    +3,  ...: 
and  negative  numbers, 

-1,    -2,    -3,  ...; 

and  by  the  introduction  of  negative  numbers,  we  can  always 
carry  out  the  mathematical  operation  of  subtraction: 

c—b  =  a, 

and,  if  b  is  greater  than  c,  a  merely  becomes  a  negative  number. 

3.  We  must  therefore  realize  that  the  negative  number  and 
the  negative  unit,  —1,  is  a  mathematical  fiction,  and  not  in 
universal  agreement  with  experience,  as  the  absolute  number 
found  in  the  operation  of  counting,  and  the  negative  number 
does  not  always  represent  an  existing  condition  in  practical 
experience. 

In  the  application  of  numbers  to  the  phenomena  of  nature, 
we  sometimes  find  conditions  where  we  can  give  the  negative 
number  a  physical  micaning,  expressing  a  relation  as  the 
reverse  to  the  positive  number;  in  other  cases  we  cannot  do 
this.  For  instance,  5  horses— 7  horses  =  —2  horses  has  no 
physical  meaning.  There  exist  no  negative  horses,  and  at  the 
best  we  could  only  express  the  relation  by  saying,  5  horses— 7 
horses  is  impossible,  2  horses  are  missing. 


THE  GENERAL  NUMBER.  5 

In  the  same  way,  an  illumination  of  5  foot-candles,  lowered 
by  3  foot-candles,  gives  an  illumination  of  2  foot-candles,  thus, 

5  foot-candles —3  foot-candles  =  2  foot-candles. 

If  it  is  tried  to  lower  the  illumination  of  5  foot-candles  by  7 
foot-candles,  it  will  be  found  impossible;  there  cannot  be  a 
negative  illumination  of  2  foot-candles;  the  limit  is  zero  illumina- 
tion, or  darkness. 

From  a  string  of  5  feet  length,  we  can  cut  off  3  feet,  leaving 

2  feet,  but  we  cannot  cut  off  7  feet,  leaving  -2  feet  of  string. 

In  these  instances,  the  negative  number  is  meaningless, 
a  mere  imaginary  mathematical  fiction. 

If  the  temperature  is  5  deg.  cent,  above  freezing,  and  falls 

3  deg.,  it  will  be  2  deg.  cent,  above  freezing.  If  it  falls  7  deg. 
it  will  be  2  deg.  cent,  below  freezing.  The  one  case  is  just  as 
real  physically,  as  the  other,  and  in  this  instance  we  may 
express  the  relation  thus: 

+5  deg.  cent.  —3  deg,  cent.  =  +2  deg.  cent., 

+  5  deg.  cent.  —7  deg.  cent.  =  —  2  deg.  cent.; 

that  is,  in  temperature  measurements  by  the  conventional 
temperature  scale,  the  negative  numbers  have  just  as  much 
physical  existence  as  the  positive  numbers. 

The  same  is  the  case  with  time,  we  may  represent  future 
time,  from  the  present  as  starting  point,  by  positive  numbers, 
and  past  time  then  will  be  represented  by  negative  numbers. 
But  we  may  equally  well  represent  past  time  by  positive  num- 
bers, and  future  times  then  appear  as  negative  numbers.  In 
this,  and  most  other  physical  applications,  the  negative  number 
thus  appears  equivalent  with  the  positive  number,  and  inter- 
changeable: we  may  choose  any  direction  as  positive,  and 
the  reverse  direction  then  is  negative.  Mathematically,  how- 
ever, a  difference  exists  between  the  positive  and  the  negative 
number;  the  positive  unit,  multiplied  by  itself,  remains  a  pos- 
itive unit,  but  the  negative  unit,  multiplied  with  itself,  does 
not  remain  a  negative  unit,  but  becomes  positive: 

(  +  1)X(  +  1)  =  (  +  1); 

(-1)X(-1)  =  (  +  1),  andnot  =(-1). 


6  ENGINEERING  MATHEMATICS. 

Starting  from  5  deg.  northern  latitude  and  going  7  deg. 
south,  brings  us  to  2  deg.  southern  latitude,  which  may  be 
expresses  thus, 

+  5  deg.  latitude  —7  deg.  latitude  =   —2  deg.  latitude. 

Therefore,  in  all  cases,  where  there  are  two  opposite  direc- 
tions, right  and  left  on  a  Hne,  north  and  south  latitude,  east 
and  west  longitude,  future  and  past,  assets  and  habilities,  etc., 
there  maj^  be  application  of  the  negative  number;  in  other  cases, 
where  there  is  only  one  kind  or  direction,  counting  horses, 
measuring  illumination,  etc.,  there  is  no  physical  meaning 
which  would  be  represented  by  a  negative  number.  There 
are  still  other  cases,  where  a  meaning  may  sometimes  be  found 
and  sometimes  not;  for  instance,  if  we  have  5  dollars  in  our 
pocket,  we  cannot  take  away  7  dollars;  if  we  have  5  dollars 
in  the  bank,  we  may  be  able  to  draw  out  7  dollars,  or  we  may 
not,  depending  on  our  credit.  In  the  first  case,  5  dollars  —7 
dollars  is  an  impossibility,  while  the  second  case  5  dollars  —7 
dollars  =  2  dollars  overdraft. 

In  any  case,  however,  w^e  must  realize  that  the  negative 
number  is  not  a  physical,  but  a  mathematical  conception, 
which  may  find  a  physical  representation,  or  may  not,  depend- 
ing on  the  physical  conditions  to  which  it  is  appUed.  The 
negative  number  thus  is  just  as  imaginary,  and  just  as  real, 
depending  on  the  case  to  which  it  is  applied,  as  the  imaginary 
number  v^  — 1,  and  the  only  difTcrence  is,  that  we  have  become 
familiar  with  the  negative  number  at  an  earlier  age,  where  we 
wer.^  less  critical,  and  thus  have  taken  it  for  gi'anted,  become 
familiar  with  it  by  use,  and  usually  do  not  realize  that  it  is 
a  mathematical  conception,  and  not  a  physical  reality.  When 
we  first  learned  it,  however,  it  was  quite  a  step  to  become 
accustomed  to  saying,  5-7= -2,  and  not  simply,  5-7  is 
impossible. 

Multiplication  and  Division. 

4.  If  we  have  a  bunch  of  4  horses,  and  another  bunch  of  4 
horses,  and  still  another  bunch  of  4  horses,  and  add  together 
the  three  bunches  of  4  horses  each,  w^e  get, 

4  horses  +4  horses +  4  horses  =  12  horses', 


THE  GENERAL  NUMBER.  7 

or,  as  we  express  it, 

3X4  horses  =  12  horses. 

The  operation  of  multiple  addition  thus  leads  to  the  next 
operation,  multiplication.  Multiplication  is  multiple  addi- 
tion, 

hXa  =  c, 

thus  means 

a  +  a  +  a-^.  .  .  (6  terms)  -=  c. 

Just  like  addition,  multiplication  can  always  be  carried 
out. 

Three  groups  of  4  horses  each,  give  12  horses.  Inversely,  if 
we  have  12  horses,  and  divide  them  into  3  equal  groups,  each 
group  contains  4  horses.  This  gives  us  the  reverse  operation 
of  multiplication,  or  diinsion,  which  is  written,  thus: 

12  horses     ^  , 

— - —  =  4  horses; 

or,  in  general, 

c 
6  =  «- 

If  we  have  a  bunch  of  12  horses,  and  divide  it  into  two  equal 
groups,  we  get  C  horses  in  each  group,  thus: 

12  horses     ^,  , 
^ =  0  horses, 

if  we  divide  into  4  equal  groups, 

12  horses 

-. =  3  horees. 

4 

If  now  we  attempt  to  divide  the  bunch  of  12  horses  into  5  equal 

groups,  we  find  we  cannot  do  it;   if  we   have  2  horses  in  each 

group,  2  horses  are  left  over;   if  we  put  3  horses  in  each  group, 

we  do  not  have  enough  to  make  5  groups;   that  is,  12  horses 

divided  by  5  is  impossible;    or,  as  we  usually  say;    12  horses 

divided  by  5  gives  2  horses  and  2  horses  left  over,  which  is 

written, 

12     ^  .    ,     ^ 

-7-  =  2,  remamder  2. 
o 


8  ENGINEERING  MATHEMATICS. 

Thus  it  is  seen  that  the  reverse  operation  of  multiplication, 
or  division,  cannot  always  he  carried  out. 

5.  If  we  have  10  apples,  and  divide  them  into  3,  we  get  3 
apples  in  each  group,  and  one  apple  left  over. 

-^-  =  3,  remainder  1, 

o 

we  may  now  cut  the  left-over  apple  into  3  equal  parts,  in  which 
case, 

In  the  same  manner,  if  we  have  12  apples,  we  can  divide 
into  5,  by  cutting  2  apples  each  into  5  equal  pieces,  and  get 
in  each  of  the  5  groups,  2  apples  and  2  pieces. 

12  1 

-^  =  2  +  2X7-=2i 
5  5 

To  be  able  to  carry  the  operation  of  division  through  for 
all  numerical  values,  makes  it  necessary  to  introduce  a  new 
unit,  smaller  than  the  original  unit,  and  derived  as  a  part  of  it. 

Thus,  if  we  divide  a  string  of  10  feet  length  into  3  equal 
parts,  each  part  contains  3  feet,  and  1  foot  is  left  over.  One 
foot  is  made  up  of  12  inches,  and  12  inches  divided  into  3  gives 
4  inches;    hence,  10  feet  divided  by  3  gives  3  feet  4  inches. 

Division  leads  us  to  a  new  form  of  numbers:  the  fraction. 

The  fraction,  however,  is  just  as  much  a  mathematical  con- 
ception, which  sometimes  may  be  applicable,  and  sometimes 
not,  as  the  negative  number.  In  the  above  instance  of  12 
horses,  divided  into  5  groups,  it  is  not  applicable. 

12  horses     ^    , 
^ =  2|  horses 

is  impossible;  we  cannot  have  fractions  of  horses,  and  what 
we  would  get  in  this  attempt  would  be  5  groups,  each  com- 
prising 2  horses  and  some  pieces  of  carcass. 

Thus,  the  mathematical  conception  of  the  fraction  is  ap- 
pUcable  to  those  physical  quantities  which  can  be  divided  into 
smaller  units,  but  is  not  applicable  to  those,  which  are  indi- 
visible, or  individuals,  as  we  usually  call  them. 


THE  GENERAL  NUMBER.  9 

Involution  and  Evolution. 

6.  If  we  have  a  product  of  several  equal  factors,  as, 
4X4X4  =  64, 
it  is  written  as,  4^  =  64 ; 

or,  in  general,  a^  =  c. 

The  operation  of  multiple  multiplication  of  equal  factors 
thus  leads  to  the  next  algebraic  operation — involution;  just  as 
the  operation  of  multiple  addition  of  equal  terms  leads  to  the 
operation  of  multiplication. 

The  operation  of  involution,  defined  as  multiple  multiplica- 
tion, requires  the  exponent  6  to  be  an  integer  number;  b  is  the 
number  of  factors. 

Thus  4-^  has  no  immediate  meaning;  it  would  by  definition 
be  4  multiplied  ( —3)  times  with  itself. 

Dividing  continuously  by  4,  we  get,  4^-^-4=4^;  4^-^4  =  4*; 
4^-^4  =  43;  q^q ^  g^j^(^[  jf  ^^jg  successive  division  by  4  is  carried 
still  further,  we  get  the  following  series: 


43     4X4X4 
4~       4 

=  4X4 

=  42 

42     4x4 

4         4 

=  4 

=  41 

41     4 
4  ~4 

=  1 

=  40 

40     1 

1 

4  ~4 

~4 

=  4-1 

^-^  =  1.4 
4       4  -^ 

1 

~4X4 

_4--i 
42 

'-'J  .4 
4       42  ■ 

1 

_4-3-l. 

^                 43' 

4X4x4 

or,  in  general, 

1. 


10  ENGINEERING  MATHEMATICS. 

Thus,  powers  with  negative  exponents,  as  a-^,  are  the 
reciprocals  of  the  same  powers  with  positive  exponents :  -^. 

7.  From  the  definition  of  involution  then  follows, 

because  a^  means  the  product  of  h  equal  factors  o,  and  a"  the 
product  of  n  equal  factors  a,  and  a^Xa^  thus  is  a  product  hav- 
ing b+n  equal  factors  a.     For  instance, 

43X42  =  (4X4X4)X(4X4)=45. 

The  question  now  arises,  whether  by  multiple  involution 
we  can  reach  any  further  mathematical  operation.     For  instance, 

(43)2  =  ?, 

may  be  written, 

(43)2  =  43x43 

=  (4X4X4)  X (4X4X4); 
=  46; 

and  in  the  same  manner, 

(a^)'*  =  a^"; 

that  is,  a  power  a^  is  raised  to  the  n***  power,  by  multiplying 
its  exponent.     Thus  also, 

that  is,  the  order  of  involution  is  immaterial. 

Therefore,  multiple  involution  leads  to  no  further  algebraic 
operations. 

8.  43  =  64; 

that  is,  the  product  of  3  equal  factors  4,  gives  64, 

Inversely,  the  problem  may  be,  to  resolve  64  into  a  product 
of  3  equal  factors.  Each  of  the  factors  then  will  be  4.  This 
reverse  operation  of  involution  is  called  evolution,  and  is  written 
thus, 

^64  =  4; 
or,  more  general, 


THE  GENERAL  NUMBER.  '  11 

Vc  thus  is  defined  as  that  number  a,  which,  raised  to  the  power 
b,  gives  c;    or,  in  other  words. 

Involution  thus  far  was  defined  only  for  integer  positive 
and  negative  exponents,  and  the  question  arises,  whether  powers 

with  fractional  exponents,  as  c^    or  c**,  have    any    meaning. 
Writing, 

\C^/    =C      ''  =c^  =  c, 

it  is  seen  that  c'>  is  that  number,  which  raised  to  the  power  6, 

gives  c;    that  is,  ct>  is  Vc,  and  the  operation  of  evolution  thus 
can  be  expressed  as  involution  with  fractional  exponent, 

c^  =  \  c, 
and 

—      /  ^  \  "      /  ^  — \  " 


or, 

and 

n 

=  {Cn)-t 

>=i 

C"> 

Obviously 

then, 

1 

n 

6 
n 

-hr- 
VC  = 

1 

-c    ^ 

1 

~  1 " 

cb 

1 

-  b    —  ■ 
A  C 

Irrational  Numbers. 

g.  Involution  with  integer  exponents,  as  4^  =  64,  can  alwa5''s 
be  carried  out.  In  many  cases,  evolution  can  also  be  carrietl 
out.     For  instance, 

^^64  =  4, 

a/T=2; 

while,  in  other  cases,  it  cannot  be  carried  out.     For  instance, 

a/J=?. 


12  ENGINEERING  MATHEMATICS. 

Attempting  to  calculate  >/J,  we  get, 

^2  =  1.4142135.  .., 

and  find,  no  matter  how  far  we  carry  the  calculation,  we  never 
come  to  an  end,  but  get  an  endless  decimal  fraction;  that  is, 
no  number  exists  in  our  system  of  numbers,  which  can  express 
'^,  but  we  can  only  approximate  it,  and  carry  the  approxima- 
tion to  any  desired  degree ;  some  such  numbers,  as  tt,  have  been 
calculated  up  to  several  hundred  decimals. 

Such  numbers  as  'v'i,  which  cannot  be  expressed  in  any 
finite  form,  but  merely  approximated,  are  called  irroiional 
numbers.  The  name  is  just  as  wrong  as  the  name  negative 
number,  or  imaginary  number.  There  is  nothing  irrational 
about  -yll.  If  we  draw  a  square,  with  1  foot  as  side,  the  length 
of  the  diagonal  is  -^'1  feet,  and  the  length  of  the  diagonal  of 
a  square  obviously  is  just  as  rational  as  the  length  of  the  sides. 

Irrational  numbers  thus  are  those  real  and  existing  numbers, 
which  cannot  be  expressed  by  an  integer,  or  a  fraction  or  finite 
decimal  fraction,  but  give  an  endless  decimal  fraction,  which 
does  not  repeat. 

Endless  decimal  fractions  frequently  are  met  when  express- 
ing common  fractions  as  decimals.  These  decimal  representa- 
tions of  common  fractions,  however,  are  'periodic  decimals, 
that  is,  the  numerical  values  periodically  repeat,  and  in  this 
respect  are  different  from  the  irrational  number,  and  can,  due 
to  their  periodic  nature,  be  converted  into  a  finite  common 
fraction.  For  ini=:tance,  2.1387387.  .  . . 
Let 

x=       2.1387387 ; 

then, 

1000j:  =  2138.7387387 

subtracting, 

999x  =  2136.0 

Hence, 

2136.6     21366     1187 77_ 

^~    999    ~  9990  ~T55"~"555" 


10.  It  is 

since, 

but  it  also  is: 

since, 


THE  GENERAL  NUMBER.  13 

Quadrature  Numbers. 

v/+r=(+2), 

(+2)X(+2)  =  (+4)-, 
v/"+4  =  (-2), 


(-2)X(-2)  =  (+4). 

Therefore,  -^4  has  two  values,  (+2)  and  (—2),  and  in 
evolution  we  thus  first  strike  the  interesting  f(^ature,  that  one 
and  the  same  operation,  with  the  same  numerical  values,  gives 
several  different  results- 

Since  all  the  positive  and  negative  numbers  are  used  up 
as  the  square  roots  of  positive  numbers,  the  question  arises. 
What  is  the  square  root  of  a  negative  number?  P'or  instance, 
a/  —4  cannot  be  —2,  as  —2  squared  gives  +4,  nor  can  it  be  +2. 

■{/— 4=  -^4X(  — l)=i-2A'-l,  and  the  question  thus  re- 
solves itself  into:  What  is  -n/^1? 

We  have  derived  the  absolute  numbers  from  experience, 
for  instance,  by  measuring  distances  on  a  hne  Fig.  4,  from  a 
starting  point  A. 

-5     -4      -3     -3     -1        0     +1      +3     +3     +4     +5 

1 1 1         Q)         I (p  I ® 1  I \ 

CAB 

Fig.  4.     Negative  and  Positive  Numbers. 

Then  we  have  seen  that  we  get  the  same  distance  from  A, 
twice,  once  toward  the  right,  once  toward  the  left,  and  this 
has  led  to  the  subdivision  of  the  numbers  into  positive  and 
negative  numbers.  Choosing  the  positive  toward  the  right, 
in  Fig.  4,  the  negative  number  would  be  toward  the  left  (or 
inversely,  choosing  the  positive  toward  the  left,  would  give 
the  negative  toward  the  right). 

If  then  we  take  a  number,  as  +2,  which  represents  a  dis- 
tance AB,  and  multiply  by  (— 1\  we  get  the  distance  AC=  —2 


14 


ENGINEERING  MATHEMATICS. 


in  opposite  direction  from  A.  Inversely,  if  we  take  AC=  —2, 
and  multiply  by  (-1),  we  get  AB= +2;  that  is,  multiplica- 
tion by  (—1)  reverses  the  direction,  turns  it  through  180  deg. 
If  we  multiply  +2  by  V-1,  we  get  +2v' -1,  a  quantity 
of  which  we  do  not  yet  know  the  meaning.  Multiplying  once 
more  by  V -I,  we  get  +2xV-lX  V-r= -2;  that  is, 
nmltiplying  a  number  +2,  twice  by  V  — 1,  gives  a  rotation  of 
180  deg.,  and  multiplication  by  \^  — 1  thus  means  rotation  by 
half  of  180  deg.;   or,  by  90  deg.,  and  +2V— 1  thus  is  the  dis- 


f^ 


OO 


v90° 


Fig.  5. 


tance  in  the  direction  rotated  90  deg.  from  +2,  or  in  quadrature 
direction  AD  in  Fig.  5,  and  such  numbers  as  +2\/  — 1  thus 
are  quadrature  numbers,  that  is,  represent  direction  not  toward 
the  right,  as  the  positive,  nor  toward  the  left,  as  the  negative 
numbers,  but  upward  or  downward^ 

For  convenience  of  writing,  V  — 1  is  usually  denoted  by 
the  letter  j. 

II.  Just  as  the  operation  of  subtraction  introduced  in  the 
negative  numbers  a  new  kind  of  numbers,  having  a  direction 
180  deg.  difTerent,  that  is,  in  opposition  to  the  positive  num- 
bers, so  the  operation  of  evolution  introduces  in  the  quadrature 
number,  as  2j,  a  new  kind  of  number,  having  a  direction  90  deg. 


THE  GENERAL  NUMBER. 


15 


different;  that  is,  at  right  angles  to  the  positive  and  the  negative 
numbers,  as  illustrated  in  Fig.  6. 

As  seen,  mathematically  the  quadrature  number  is  just  as 
real  as  the  negative,  physically  sometimes  the  negative  number 
has  a  meaning — if  two  opposite  directions  exist — ;  sometimes  it 
has  no  meaning — where  one  direction  only  exists.  Thus  also 
the  quadrature  number  sometimes  has  a  physical  meaning,  in 
those  cases  where  four  directions  exist,  and  has  no  meaning, 
in  those  physical  problems  where  only  two  directions    exist. 


+  4/ 
+  3;- 


-4     -3     -3     -1     0 


— I — I — I — ^ 

+1     +2     +3     r4 
-3 

+-3y 


-J 

Fig.  6. 


For  instance,  in  problems  dealing  with  plain  geometry,  as  in 
electrical  engineering  when  discussing  alternating  current 
vectors  in  the  plane,  the  quadrature  numbers  represent  the 
vertical,  the  ordinary  numbers  the  horizontal  direction,  and  then 
the  one  horizontal  direction  is  positive,  the  other  negative,  and 
in  the  same  manner  the  one  vertical  direction  is  positive,  the 
other  negative.  Usually  positive  is  chosen  to  the  right  and 
upward,  negative  to  the  left  and  downward,  as  indicated  in 
Fig.  6.  In  other  problems,  as  when  dealing  with  time,  which 
has  only  two  directions,  past  and  future,  the  quath-ature  num- 
bers are  not  applicable,  but  only  the  positive  and  negative 


16 


ENGINEERING  MATHEMATICS. 


numbers.  In  still  other  problems,  as  when  deahng  with  illumi- 
nation, or  with  individuals,  the  negative  numbers  are  not 
applicable,  but  only  the  absolute  or  positive  numbers. 

Just  as  multiplication  by  the  negative  unit  (—1)  means 
rotation  by  180  deg.,  or  reverse  of  direction,  so  multiplication 
by  the  quadrature  unit,  j,  means  rotation  by  90  deg.,  or  change 
from  the  horizontal  to  the  vertical  direction,  and  inversely. 


General    Numbers. 


12.  By  the  positive  and  negative  numbers,  all  the  points  of 
a  line  could  be  represented  numerically  as  distances  from  a 
chosen  point  A. 


Fig.  7.     Simple  Vector  Diagram. 

By  the  addition  of  the  quadrature  numbers,  all  points  of 
the  entire  plane  can  now  be  represented  as  distances  from 
chosen  coordinate  axes  x  and  y,  that  is,  any  point  P  of  the 
plane,  Fig.  7,  has  a  horizontal  distance,  0B=  +3,  and  a 
vertical  distance,  BP=  -\-2j,  and  therefore  is  given  by  a 
combination  of  the  distances,  0B=  +3  and  BP=  +2;.  For 
convenience,  the  act  of  combining  two  such  distances  in  quad- 
rature with  each  other  can  be  expressed  by  the  plus  sign, 
and  the  result  of  combination  thereby  expressed  by  OB  \-BP 
=  3+2/. 


THE  GENERAL  NUMBER. 


17 


Such  a  combination  of  an  ordinary  number  and  a  quadra- 
ture number  is  called  a  general  number  or  a  complex  quantity. 

The  quadrature  number  jb  thus  enormously  extends  the 
field  of  usefulness  of  algebra,  by  affording  a  numerical  repre- 
sentation of  two-dimensional  systems,  as  the  j^lane,  by  the 
general  number  a  4-/6.  They  are  especially  useful  and  impor- 
tant in  electrical  engineering,  as  most  problems  of  alternating 
currents  lead  to  vector  representations  in  the  plane,  and  there- 
fore can  be  represented  by  the  general  number  a+jh;  that  is, 
the  ccmbinaticn  of  the  ordinary  number  or  horizontal  distance 
a,  and  the  quadrature  number  or  vertical  distance  jh. 


Fig.  S.     Vector  Diagram. 

Analytically,  points  in  the  plane  are  represented  by  their 
two  coordinates:  the  horizontal  coordinate,  or  abscissa  x,  and 
the  vertical  coordinate,  or  ordinate  y.  Algebraically,  in  the 
general  number  a+jb  both  coordinates  are  combined,  a  being 
the  X  coordinate,  jb  the  y  coordinate. 

Thus  in  Fig.  8,  coordinates  of  the  points  are. 

Pi:    x=+S,     y=+2  P.:    a-= +3     y= -2, 

P3:    x=-3,     y=+2  P,:    x=-3     y= -2, 

and  the  points  are  located  in  the  plane  by  the  numbers: 
Pi=3+2y    P2  =  3-2j    P3=-3+2y    P4=-3-2y 


18  ENGINEERING  MATHEMATICS. 

13.  Since  already  the  square  root  of  negative  numbers  has 
extended  the  system  of  numbers  by  giving  the  quadrature 
number,  the  question  arises  whether  still  further  extensions 
of  the  system  of  numbers  would  result  from  higher  roots  of 
negative  quantities. 

For  instance, 

The  meaning  of   V^l  we  find  in  the  same  manner  as  that 

of  ^^. 

A  positive  number  a  may  be  represented  on  the  horizontal 
axis  as  P. 

Multiplying  a  by  a'— 1  gives  a-^  — 1,  whose  meaning  we  do 
not  yet  know.  Multiplying  again  and  again  by  "^  —  1,  we  get,  after 
four  multiplications,  a(-^— 1)^=  —a;  that  is,  in  four  steps  we 
have  been  carried  from  a  to    —a,  a  rotation  of  180  deg.,  and 

'^l  thus  means  a  rotation  of ——  =  45  deg.,  therefore,  a'>l —1 

is  the  point  Pi  in  Fig.  9,  at  distance  a  from  the  coordinate 
center,  and  under  angle  45  deg.,  which  has  the  coordinates, 

x  =  — =  and  y  =  — =/;  or,  is  represented  by  the  general  number, 

V2  V2 

P   -a^ 

■>! -1,  however,  may  also  mean  a  rotation  by  135  deg.  to  P2, 
since  this,  repeated  four  times,  gives  4x135  =  540  deg., 
or  the  same  as  180  deg.,  or  it  may  mean  a  rotation  by  225  deg. 
or  by  315  deg.  Thus  four  points  exist,  which  represent  a-yj  -1; 
the  points: 

V2  v2 

\  2  v2 

Therefore,  ^ —1  is  still  a  general  number,  consisting  of  an 
ordinary  and  a  quadrature  number,  and  thus  does  not  extend 
our  system  of  numbers  any  further. 


THE  GENERAL  NUMBER. 


19 


In  the  same  manner,  -^  +  1  can  be  found;  it  is  that  number, 

which,  multiplied  n  times  with  itself,  gives  +1.     Thus  it  repre- 

.       ,      360    , 
sents  a  rotation  by  —  dcg.,  or  any  multiple  thereof;   that  is, 

,•     .     .  360     ,  '  360 

the  X  coordmate  is  cos  qX — ,  the  y  coordinate  sin  g-X — , 

n  n 

and, 

ny—r  360        .    .  360 

V  +l  =  cos  </X +  7  sin  oX  — , 


where  q  is  any  integer  number. 


Fig.  9.     Vector  Diagram  a-^—  1. 


There  are  therefore  n  different  values  of  a  "^ +  1,  which  lie 
equidistant  on  a  circle  with  radius  1,  as  shown  for  7i  =  9  in 
Fig.  10. 

14.  In  the  operation  of  addition,  a  +  6  =  c,  the  problem  is, 
a  and  h  being  given,-  to  find  c. 

The  terms  of  addition,  a  and  6,  are  interchangeable,  or 
equivalent,  thus :  a  +  6  =  6  +  a,  and  addition  therefore  has  only 
one  reverse  operation,  subtraction;  c  and  h  being  given,  a  is 
found,  thus:  a  =  c—b,  and  c  and  a  being  given,  b  is  found,  thus: 
b=c—a.      Either  leads   to   the   same   operation — subtraction. 

The   same   is   the   case   in   multiplication;     aXb  =  c.    The 


QO 


ENGINEERING  MA  THEM  A  TICS. 


factors  a  and  b  are  interchangeable  or  equivalent;    aXb  =  bXa 

.  .  c  c 

and  the  reverse  operation,  division,  a  =  ^  is  the  same  as  b  =  —. 

0  a 

In  involution,  however,  a^  =  c,  the  two  numbers  a  and  b 
are  not  interchangeable,  and  a^  is  not  equal  to  6".  For  instance 
43  =  64  and  34  =  81. 

Therefore,  involution  has  two  reverse  operations: 

(a)  c  and  b  given,  a  to  be  found, 


b  — 
a  —  V  c ; 


or  evolution. 


Fig.  10.     Points  Determined  by  V+l. 


(6)  c  and  a  given,  b  to  be  found, 
or,  logarithmation. 


6  =  loga  c; 


Logarithmation. 

15.  Logarithmation  thus  is  one  of  the  reverse  operations 
of  involution,  and  the  logarithm  is  the  exponent  of  involution. 

Thus  a  logaritjimic  expression  may  be  changed  to  an  ex- 
ponential, and  inversely,  and  the  laws  of  logarithmation  are 
the  laws,  which  the  exponents  obey  in  involution. 

1.  Powers  of  equal  base  are  multiplied  by  adding  the 
exponents:     a^'Xa'^^a^'^'*.      Therefore,    the    logarithm    of    a 


THE  GENERAL  NUMBER.  21 

product  is  the  sum  of  the  logarithms  of  the  factors,  thus  loga  cXd 
=  loga  c+loga  d. 

2.  A  power  is  raised  to  a  power  by  multiplying  the  exponents: 

Therefore  the  logarithm  of  a  power  is  the  exponent  times 
the  logarithm  of  the  base,  or,  the  number  under  the  logarithm 
is  raised  to  the  power  n,  by  multiplying  the  logarithm  by  n: 

loga   C''=n  loga   C, 

loga  1=0,  because  a9  =  l.  If  the  base  a  >  1,  loga  c  is  positive, 
if  c>l,  and  is  negative,  if  c<l,  but  >0.  The  reverse  is  the 
case,  if  a<l.  Thus,  the  logarithm  traverses  all  positive  and 
negative  values  for  the  positive  values  of  c,  and  the  logarithm 
of  a  negative  number  thus  can  be  neither  positive  nor  negative. 

loga  (— c)=loga  c+loga  (" 1),  and  the  question  of  finding 
the  logarithms  of  negative  numbers  thus  resolves  itself  into 
finding  the  value  of  loga  (  —  !)• 

There  are  two  standard  systems  of  logarithms  one  with 
the  base  £  =  2.71828.  .  .*,  and  the  other  with  the  base  10  is 
used,  the  former  in  algebraic,  the  latter  in  numerical  calcula- 
tions. Logarithms  of  any  base  a  can  easily  be  reduced  to  any 
other  base. 

For  instance,  to  reduce  6  =  loga  c  to  the  base  10:  6  =  loga  c 
means,  in  the  form  of  involution:  a^=c.  Taking  the  logarithm 
hereof  gives,  b  logio  a  =  logio  c,  hence, 

,      logioc  ,  logioc 

0  =  1 ;     or   loga  c  =  i . 

logio  a  *=  logio  a 

Thus,  regarding  the  logarithms  of  negative  numbers,  we  need 
to  consider  only  logio  (  —  1)  or  log^  (  -l^)- 

If  P  =  log,  (-1),  then  £'^=  -1, 

and  since,  as  will  be  seen  in  Chapter  II, 

e'^  =  cos  x  +  j  sin  x, 
it  follows  that, 

cos  x  +  y  sin  X  =  —  1, 

*  Regarding  e,  see  Chapter  II,  p.  71. 


22  ENGINEERING  MATHEMATICS. 

Hence,  a;  =  7r,  or  an  odd  multiple  thereof,  and 

log£(-l)=y;:(2n  +  l), 

where  n  is  any  integer  number. 

Thus  logarithmation  also  leads  to  the  quadrature  number 
/,  but  to  no  further  extension  of  the  system  of  numbers. 

Quaternions. 

i6.  Addition  and  subtraction,  multiplication  and  division, 
involution  and  evolution  and  logarithmation  thus  represent  all 
the  algebraic  operations,  and  the  system  of  numbers  in  which 
all  these  operations  can  be  carried  out  under  all  conditions 
is  that  of  the  general  number,  a  +  jb,  comprising  the  ordinary 
number  a  and  the  quadrature  number  jh.  The  number  a  as 
well  as  h  may  be  positive  or  negative,  may  be  integer,  fraction 
or  irrational. 

Since  by  the  introduction  of  the  quadrature  number  jh, 
the  application  of  tbe  system  of  numbers  was  extended  from  the 
line,  or  more  general,  one-dimensional  quantity,  to  the  plane, 
or  the  two-dimensional  quantity,  the  question  arises,  whether 
the  system  of  numbers  could  be  still  further  extended,  into 
three  dimensions,  so  as  to  represent  space  geometry.  While 
in  electrical  engineering  most  problems  lead  only  to  plain 
figures,  vector  diagrams  in  the  plane,  occasionally  space  figures 
would  be  advantageous  if  they  could  be  expressed  algc^bra- 
ically.  Especially  in  mechanics  this  would  be  of  importance 
when  dealing  with  forces  as  vectors  in  space. 

In  the  quaternion  calculus  methods  have  been  devised  to 
deal  with  space  problems.  The  quaternion  calculus,  however, 
has  not  yet  found  an  engineering  application  comparable  with 
that  of  the  general  number,  or,  as  it  is  frequently  called,  the 
complex  quantity.  The  reason  is  that  the  quaternion  is  not 
an  algebraic  quantity,  and  the  laws  of  algebra  do  not  uniformly 
apply  to  it. 

17.  With  the  rectangular  coordinate  system  in  the  plane. 
Fig.  11,  the  X  axis  may  represent  the  ordinary  numbers,  the  y 
axis  the  quadrature  numbers,  and  multiphcation  by  j  =  V— 1 
represents  rotation  by  90  deg.     For  instance,  if  Pi  is  a  point 


THE  GENERAL  NUMBER. 


23 


a+jb  =  S+2j,    the    point  P2,  90  deg.  away    from   Pi,  would 
be: 

^2  =yPi  -j{a+jb)=j(S  +2])  =  -2+3/, 

To  extend  into  space,  we  have  to  add  the  third  or  z  axis, 
as  shown  in  perspective  in  Fig.  12.  Rotation  in  the  plane  xy, 
by  90  deg.,  in  the  direction  +x  to  +y,  then  means  multiplica- 
tion by  /.  In  the  same  manner,  rotation  in  the  yz  plane,  by 
90  deg.,  from  +y  to  +z,  would  be  represented  by  multiplica- 


>  + 


Fig.  11.     Vectors  in  a  Plane. 

tion  with  h,  and  rotation  by  90  deg.  in  the  zx  plane,  from  +z 
to  +x  would  be  presented  by  k,  as  indicated  in  Fig.  12. 

All  three  of  these  rotors,  j,  h,  k,  would  be  V  — 1,  since  each, 
apphed  twice,  reverses  the  direction,  that  is,  represents  multi- 
phcation  by  (—1). 

As  seen  in  Fig.  12,  starting  from  +z,  and  going  to  +y, 
then  to  +z,  and  then  to  +x,  means  successive  multipHcation 
by  /,  h  and  k,  and  since  we  come  back  to  the  starting  point,  the 
total  operation  produces  no  change,  that  is,  represents  mul- 
tiplication by  (  +  1).     Hence,  it  must  be, 

jhk=  +1. 


24 


ENGINEERING  MATHEMATICS. 


Algebraically  this  is  not  possible,  since  each  of  the  three  quan- 
tities is   V-1,   and    \/-lxV-lxV-l=  -V-1,  and  not 

(  +  1). 


>+x 


Fig.  12.     Vectors  in  Space,  jhk—  +1. 

If  we  now  proceed  again  from  x,  in  positive  rotation,  but 
first  turn  in  the  xz  plane,  we  reach  by  multiplication  with  A: 
the  negative  z  axis,  —z,  as  seen  in  Fig.  13.     Further  multipHca- 


+z 


rz 


-►+« 


-y 

IiG.  13.     Vectors  in  Space,  hhj=  —  1. 


tion  by  h  brings  us  to  -^y,  and  multiplication  by  j  to  —x,  and 
in  this  case  the  result  of    the  three  successive  rotations  by 


THE  GENERAL  NUMBER.  25 

90  deg.,  in  the  same  direction  as  in  Fig.  12,   but  in  a  different 
order,  is  a  reverse;    that  is,  represents  (—1).     Therefore, 

khj=  -1, 
and  hence, 

jhk=  —khj. 

Thus,  in  vector  analysis  of  space,  we  see  that  the  fundamental 
law  of  algebra, 

aXb  =  bXa, 

does  not  apply,  and  the  order  of  the  factors  of  a  product  is 
not  immaterial,  but  by  changing  the  order  of  the  factors  of  the 
product  jhk,  its  sign  was  reversed.  Thus  common  factors  can- 
not be  canceled  as  in  algebra;  for  instance,  if  in  the  correct  ex- 
pression, jhk  =  —  khj,  we  should  cancel  by  /,  h  and  k,  as  could  be 
done  in  algebra,  we  would  get  + 1  =  —1,  which  is  obviously  wrong. 
For  this  reason  all  the  mechanisms  devised  for  vector  analysis 
in  space  have  proven  more  difficult  in  their  application,  and 
have  not  yet  been  used  to  any  great  extent  in  engineering 
practice. 

B.  ALGEBRA  OF  THE  GENERAL  NUMBER,  OR  COMPLEX 

QUANTITY. 

Rectangular  and  Polar  Coordinates. 

i8.  The  general  number,  or  complex  quantity,  a  4-/6,  is 
the  most  general  expression  to  which  the  laws  of  algebra  apply. 
It  therefore  can  be  handled  in  the  same  manner  and  under 
the  same  rules  as  the  ordinary  number  of  elementary  arithmetic. 
The  only  feature  which  must  be  kept  in  mind  is  that  j^  =  —  1,  and 
where  in  multiplication  or  other  operations  p  occurs,  it  is  re- 
placed by  its  value,  —1.     Thus,  for  instance, 

(a + jb)  (c  +  jd)  =ac+  jad  +  jbc  +  fbd 
=  ac-\-  jad  +  jbc  —  bd 
=  (ac  —bd)  +j{ad  +  bc). 

Herefrom  it  follows  that  all  the  higher    powers  of  ;  can  be 
eUminated,  thus: 

j^ = i,    j^=  - 1,  f  =  -h  y*  =  + 1 ; 
f  =  +j,  f  =  -1,  f  =  -j,  f =+1; 

y^=  +j,  .  .  .    etc. 


26  ENGINEERING  MATHEMATICS. 

In  distinction  from  the  general  number  or  complex  quantity, 
the  ordinary  numbers,  +a  and  —a,  arc  occasionally  called 
scalars,  or  real  numbers.  The  general  number  thus  consists 
of  the  combination  of  a  scalar  or  real  number  and  a  quadrature 
number,  or  imaginary  number. 

Since  a  quadrature  number  cannot  be  equal  to  an  ordinary 
number  it  follows  that,  if  two  general  numbers  are  equal, 
their  real  components  or  ordinary  numbers,  as  well  as  their 
quadrature  numbers  or  imaginary  components  must  be  equal, 
thus,  if 

a  +  jb==c+]'d, 
then, 

a  =  c    and     b  =  d. 

Every  equation  with  general  numbers  thus  can  be  resolved 
into  two  equations,  one  containing  only  the  ordinary  numbers, 
the  other  only  the  quadrature  numbers.     For  instance,  if 

x+jy  =  5-Sj, 
then, 

x  =  5     and     y=  —3. 

19.  The  best  way  of  getting  a  conception  of  the  general 
number,  and  the  algebraic  operations  with  it,  is  to  consider 
the  general  number  as  representing  a  point  in  the  plane.  Thus 
the  general  number  0  4-/6  =  6+2.5/  may  be  considered  as 
representing  a  point  P,  in  Fig.  14,  which  has  the  horizontal 
distance  from  the  y  axis,  OA  =  BP  =  a  =  G,  and  the  vertical 
distance  from  the  x  axis,  OB  =  AP  =  h  =  2.5. 

The  total  distance  of  the  point  P  from  the  coordinate  center 
0  then  is 

0P  =  ^0A2+AP2 


=  Va2  +  62  =  V62  +  2 .52  =  6.5, 

and  the  angle,  which  this  distance  OP  makes  with  the  x  axis, 
is  given  by 

AP 


tan  6  = ': 

OA 


¥--• 


THE  GENERAL  NUMBER. 


27 


Instead  of  representing  the  general  number  by  the  two 
components,  a  and  b,  in  the  form  a+jb,  it  can  also  be  repre- 
sented by  the  two  quantities:  the  distance  of  the  point  P  from 
the  center  0, 


and  the  angle  between  this  distance  and  the  x  axis, 


tan6'=- 


a 


H 1 H 


Fic.  14.     Rectangular  and  Polar  Coordinates. 

Then  referring  to  Fig.  14, 

a  =  ccos^     and     b  =  c  sin  d, 

and  the  general  number  a+jb  thus  can  also  be  written  in  the 
form, 

c(cos  6  +ysin  6). 

The  form  a+jb  expresses  the  general  number  by  its 
rectangular  components  a  and  b,  and  corresponds  to  the  rect- 
angular coordinates  of  analytic  geometry;  a  is  the  x  coordinate, 
b  the  y  coordinate. 

The  form  c{cos  d  +  j  sin  6)  expresses  the  general  number  by 
what  may  be  called  its  polar  components,  the  radius  c  and  the 


28  ENGINEERING  MATHEMATICS. 

angle  d,  and  corresponds  to  the  polar  coordinates  of  analjrtic 
geometry,  c  is  frequently  called  the  radius  vector  or  scalar, 
6  the  phase  angle  of  the  general  number. 

While  usually  the  rectangular  form  a+jh  is  more  con- 
venient, sometimes  the  polar  form  c(cos  ^  +y  sin  ^)  is  preferable, 
and  transformation  from  one  form  to  the  other  therefore  fre- 
quently applied. 

Addition  and  Subtraction. 

20.  If  ai +y6i  =  6+2.5  J  is  represented  by  the  point  Pi; 
this  point  is  reached  by  going  the  horizontal  distance  ai  =  6 
and  the  vertical  distance  hi  =2.5.  If  02+^62  =  3+4/  is  repre- 
sented by  the  point  P2,  this  point  is  reached  by  going  the 
horizontal  distance  02  =  8  and  the  vertical  distance  &2  =  4. 

The  sum  of  the  two  general  numbers  (ai  +jbi)  +  (o2+j62)  = 
(6+2.5/) +  (3+4/),  then  is  given  by  point  Pq,  which  is  reached 
by  going  a  horizontal  distance  equal  to  the  sum  of  the  hor- 
izontal distances  of  Pi  and  P2:  ao  =  ai +a2  =  6+3  =  9,  and  a 
vertical  distance  equal  to  the  sum  of  the  vertical  distances  of 
Pi  and  P2:  6o  =  &i +&2  =  2.5+4  =  6.5,  hence,  is  given  by  the 
general  number 

ao  +/&o  =  (fli  +^2)  +j(bi  +&2) 
=  9  +  6.5/. 

Geometrically,  point  Pq  is  derived  from  points  Pi  and  P2^ 
by  the  diagonal  OPq  of  the  parallelogram  OPiPqPo,  constructed 
with  OPi  and  OP2  as  sides,  as  seen  in  Fig,  15. 

Herefrom  it  follows  that  addition  of  general  numbers 
represents  geometrical  combination  by  the  parallelogram  law. 

Inversely,  if  Pq  represents  the  number 

ao+/6o  =  9  +  6.5/, 

and  Pi  represents  the  number 

ai+/6i=6+2.5/, 

the  difference  of  these  numbers  will  be  represented  by  a  point 
P2,  which  is  reached  by  going  the  difference  of  the  horizontal 


THE  GENERAL  NUMBER. 


29 


distances  and  of  the  vertical  distances  of  the   points  Pq  and 
Pi-     P2  thus  is  represented  by 


and 


a2  =  ao  — ai  =  9— 6  =  3, 
62  =  60-61=6.5-2.5  =  4. 


Therefore,  the  difference  of  the  two  general  numbers  (ao+jho) 
and  (oi  +jbi)  is  given  by  the  general  number: 


as  seen  in  Fir.  15. 


a2+jb2  =  (aQ-ai)+i{bo  -61) 
=  3+4y, 


Fig.  15.     Addition  and  Subtraction  of  Vectors. 

This  difference  a2+jb2  is  represented  by  one  side  OF2  of 
the  parallelogram  OP1P0P2,  which  has  OPi  as  the  other  side, 
and  OPq  as  the  diagonal. 

Subtraction  of  general  numbers  thus  geometrically  represents 
the  resolution  of  a  vector  OPq  into  two  components  OPi  and 
OP2,  by  the  parallelogram  law. 

Herein  lies  the  main  advantage  of  the  use  of  the  general 
number  in  engineering  calculation :  If  the  vectors  are  represented 
by  general  numbers  (complex  quantities),  combination  and 
resolution  of  vectors  by  the  parallelogram  law  is  carried  out  by 


30  ENGINEERING  MATHEMATICS. 

simple  addition  or  subtraction  of  their  general  numerical  values, 
that  is,  by  the  simplest  operation  of  algebra. 

21.  General  numbers  are  usually  denoted  by  capitals,  and 
their  rectangular  components,  the  ordinary  number  and  the 
quadrature  number,  by  small  letters,  thus: 

A  =  ai+ja2; 

the  distance  of  the  point  which  represents  the  general  number  A 
from  the  coordinate  center  is  called  the  absolute  value,  radius 
or  scalar  of  the  general  number  or  complex  quantity.  It  is 
the  vector  a  in  the  polar  representation  of  the  general  number: 

A  =  a(cos  d+j  sin  6), 


and  is  given  by  o=  Vat^  +  a-r. 

The  absolute  value,  or  scalar,  of  the  general  number  is  usually 
also  denoted  by  small  letters,  but  sometimes  by  capitals,  and 
in  the  latter  case  it  is  distinguished  from  the  general  number  by 
using  a  different  type  for  the  latter,  or  underhning  or  dotting 
it,  thus: 

A  =  ai+ja2;         or     A  =  ai+ja2,0T  A  =  ai+]'a2 

or       A  =  ai+  jao ;  or    A  =  Oi  +  ]'a2 


a  =  Va{^  +  a-r ;     or     A  =  \^a{'-  +  02^, 

and  ai +/a2  =  a(cos  ^  +  /sin  ^); 

or  tti  +  jao  =  A  (cos  ^  + ;  sin  d). 

22.  The  absolute  value,  or  scalar,  of  a  general  number  is 
always  an  absolute  number,  and  positive,  that  is,  the  sign  of  the 
rectangular  component  is  represented  in  the  angle  d.  Thus 
referring  to  Fig.  16, 

.4  =ai+ya2  =  4+3/; 
gives,  a  =  \'^ar+a2-  =  5; 

tan  ^  =  1=0.75; 
^  =  37  deg.; 
and  .4=5  (cos  37  deg.  +/sin  37  deg). 


The  expression 


gives 


THE  GENERAL  NUMBER. 


A  =  ai+ja2  =  i-Sj 


a  =  Vai^  4-  az^  =  5; 


31 


tan  ^=--  =  -0.75; 
4 

^=-37deg.;     or     =180-37  =  143  deg. 


Fig.  16.     Representation  of  General  Numbers. 


Which  of  the  two  values  of  6  is  the  correct  one  is  seen  from 
the  condition  ai  =  acos^.  As  ai  is  positive,  +4,  it  follows 
that  cos  6  must  be  positive;  cos  (—37  deg.)  is  positive,  cos  143 
deg.  is  negative:  hence  the  former  value  is  correct: 

A  -5{cos(  -37  deg.)  +/  sin(  -37  deg.)} 
=  5 (cos  37  deg.  — ;  sin  37  deg.). 

■''Two  such  general    numbers  as    (4  4-3/)   and     (4—3;),   or, 
in  general, 

(a+jb)     and     (a—jh), 

are  called  conjugate   numbers.     Their   product  is  an   ordinary 
and  not  a  general  number,  thus :   (a  +  jh)  (a  -jb)  =  a^  +  b^. 


32  ENGINEERING  MATHEMATICS. 

The  expression 

A  =  ai+ja2=  —4+3/ 
gives 


3 

tan^=  --=  -0.75; 

e=  -Zl  dog.     or     =  180  -37  =  143  cleg. ; 

but  since  a\=a  cos  0  is  negative,  —4,  cos  0  must  be  negative, 
hence,  d  =  143  deg.  is  the  correct  vakie,  and 

^=5(cos  143  deg.  +/sin  143  deg.) 
=  5(— cos  37  deg. +j  sin  37  deg.) 

The  expression 

4=oi+y«2=  -4-3y 

gives 


a=  V'ai2+a2^  =  5; 

^  =  37  dog.;     or     =180+37  =  217  deg.; 

but  since  ai=a  cos  d  is  negative,   —4,  cos  d  must  be  negative, 
hence  ^  =  217  deg.  is  the  correct  value,  and, 

4  =  5  (cos  217  deg.  +/  sin  217  deg.) 
=  5(  —  cos  37  deg.  —/sin  37  deg.) 

The  four  general  numbers,  +4+3/,  +4—3/,  —4+3/,  and 
—4  —3/,  have  the  same  absolute  value,  5,  and  in  their  repre- 
sentations as  points  in  a  plane  have  symmetrical  locations  in 
the  four  quadrants,  as  shown  in  Fig.  16. 

As  the  general   number   A  =  ai+/a2   finds   its  main  use  in 
representing  vectors  in  the  plane,  it  very  frequently  is  called 
a  vector  quantity,  and  the  algebra  of  the  general  number  ia  m 
spoken  of  as  vector  analysis. 

Since  the  general  numbers  4  =  ^1+  y^2  can  be  made  to 
represent  the  points  of  a  plane,  they  also  may  be  called  plane 
numbers,  while  the  positive  and  negative  numbers,  +a  and— a, 


THE  GENERAL   NUMBER.  33 

may  be  called  the  linear  numbers,  as  they  represent  the  points 
of  a  line. 

Example :  Steam  Path  in  a  Turbine. 

23.  As  an  example  of  a  simple  operation  with  general  num- 
bers one  may  calculate  the  steam  path  in  a  two-wheel  stage 
of  an  impulse  steam  turbine. 


■i-y 


I 


-)»)»))))- 


>  +a; 


Fig.  17.     Path  of  Steam  in  a  Two-wheel  Stage  of  an  Impulse  Turbine. 

Let  Fig.  17  represent  diagrammatically  a  tangential  section 
through  the  bucket  rings  of  the  turbine  wheels.  Wi  and  14^2 
are  the  two  revolving  wheels,  moving  in  the  direction  indicated 
by  the  arrows,  with  the  velocity  s  =  400  feet  per  sec.  /  are 
the  stationary  intermediate  buckets,  which  turn  the  exhaust 
steam  from  the  first  bucket  wheel  Wi,  back  into  the  direction 
required  to  impinge  on  the  second  bucket  wheel  W2.  The 
steam  jet  issues  from  the  expansion  nozzle  at  the  speed  So  =  2200 


34 


ENGINEERING  MA  THEM  A  TICS. 


feet  per  sec,  and  under  the  angle  ^o  =  20  deg.,  against"  the  first 
bucket  wheel  W^. 

The  exhaust  angles  of  the  three  successive  rows  of  buckets, 
Wi,  /,  and  W2,  are  respectively  24  deg.,  30  deg.  and  45  deg. 
These  angles  are  calculated  from  the  section  of  the  bucket 
exit  required  to  pass  the  steam  at  its  momentary  velocity, 
and  from  the  height  of  the  passage  required  to  give  no  steam 
eddies,  in  a  manner  which  is  of  no  interest  here. 

As  friction  coefficient  in  the  bucket  passages  may  be  assumed 
Ay  =  0.12;  that  is,  the  exit  velocity  is  1  — Ay =0.88  of  the  entrance 
velocity  of  the  steam  in  the  buckets. 


Fig.  18.     Vector  Diagram  of  Velocities  of  Steam  in  Turbine. 

Choosing  then  as  x-axis  the  direction  of  the  tangential 
velocity  of  the  turbine  wheels,  as  y-axis  the  axial  direction, 
the  velocity  of  the  steam  supply  from  the  expansion  nozzle  is 
represented  in  Fig.  18  by  a  vector  OSq  of  length  so  =  2200  feet 
per  sec,  making  an  angle  ^o  =  20  deg.  with  the  j-axis;  hence, 
can  be  expressed  by  the  general  number  or  vector  quantity: 

-^o  =  So  (cos  ^0  +]  sin  ^0) 

=  2200  (cos  20  deg.  +/  sin  20  deg.) 
=  2070  +  750/ ft.  per  sec 

The  velocity  of  the  turbine  wheel  Wi  is  s  =  400  feet  per  second, 
and  represented  in  Fig.  18  by  the  vector  OS,  in  horizontal 
direction. 


THE  GENERAL  NUMBER.  35 

The  relative  velocity  with  which  the  steam  enters  the  bucket 
passage  of  the  first  turbine  wheel  Wx  thus  is : 

=  (2070 +750/) -400 
=  1670+750/ ft.  per  sec. 

This  vector  is  shown  as  OSi  in  Fig.  18. 
The  angle  Oi,  under  which  the  steam  enters  the   bucket 
passage  thus  is  given  by 

750 
tan  dx  =  — ^  =  0.450,     as    Ox  =  24.3  deg. 

This  angle  thus  has  to  be  given  to  the  front  edge  of  the 
buckets  of  the  turbine  wheel  TFi. 

The  absolute  value  of  the  relative  velocity  of  steam  jet 
and  turbine  wheel  Wx,  at  the  entrance  into  the  bucket  passage, 
is 


Si  =  \/16702  + 7502  =  1830  ft.  per  sec. 

In  traversing  the  bucket  passages  the  steam  velocity  de- 
creases by  friction  etc.,  from  the  entrance  value  Sx  to  the 
exit  value 

S2  =  si(l- Ay)  =  1830X0.88  =  1610  ft.  per  sec, 

and  since  the  exit  angle  of  the  bucket  passage  has  been  chosen 
as  ^2  =  24  deg.,  the  relative  velocity  with  which  the  steam 
leaves  the  first  bucket  wheel  Wx  is  represented  by  a  vector 
OS^  in  Fig.  18,  of  length  S2  =  1610,  under  angle  24  deg.  The 
steam  leaves  the  first  wheel  in  backward  direction,  as  seen  in 
Fig.  17,  and  24  deg.  thus  is  the  angle  between  the  steam  jet 
and  the  negative  x-axis;  hence,  ^2  =  180—24  =  156  deg.  is  the 
vector  angle.  The  relative  steam  velocity  at  the  exit  from 
wheel  Wx  can  thus  be  represented  by  the  vector  quantity 

>S2  =  S2(cos  ^2+/ sin  ^2) 

=  1610  (cos  156  deg.  +;'  sin  156  deg.) 
=  -1470  +  655/. 

Since  the  velocity  of  the  turbine  wheel  Wx  is  .s  =  400,  the 
velocity  of  the  steam  in  space,  after  leaving  the  first  turbine 


36  ENGINEERING  MATHEMATICS. 

wheel,  that  is,  the  velocity  with  which  the  steam  enters  the 
intermediate  /,  is 

^3  =  ^2+5 

=  (-1470 +655/) +400 
=  -1070  +  655/, 

and  is  represented  by  vector  OS;i  in  Fig.  18, 
The  direction  of  this  steam  jet  is  given  by 

055 
tan^3=-j(3^=-0.613, 

as 

(?3  =  -31.6  dcg. ;     or,     180  -31.6  =  148.4  deg. 

The  latter  value  is  correct,  as  cos  ^3  is  negative,  and  sin  6^  is 
positive. 

The  steam  jet  thus  enters  the  intermediate  under  the  angle 
of  148.4  deg. ;  that  is,  the  angle  180  -148.4  =  31.6  deg.  in  opposite 
direction.  The  buckets  of  the  intermediate  /  thus  must  be 
curved  in  reverse  direction  to  those  of  the  wheel  TFi,  and  must 
be  given  the  angle  31.6  dcg.  at  their  front  edge. 

The  absolute  value  of  the  entrance  velocity  into  the  inter- 
mediate /  is 


S3  =  \/l0702  +  6552  =  1255  ft.  per  sec. 

In  passing  through  the  bucket  passages,  this  velocity  de- 
creases by  friction,  to  the  value : 

S4  =  83(1 -A-^)  =  1255X0.88  =  1105  ft.  per  sec, 

and  since  the  exit  edge  of  the  intermediate  is  given  the  angle: 
^4  =  30  deg.,  the  exit  velocity  of  the  steam  from  the  intermediate 
is  represented  by  the  vector  0*S4  in  Fig.  18,  of  length  6-4  =  1105, 
and  angle  ^4  =  30  deg.,  hence, 

,§4  =  1105  (cos  30  deg.  +/  sin  30  deg.) 
=  955  +  550/  ft.  per  sec. 

This  is  the  velocity  with  which  the  steam    jet  impinges 
on  the  second  turbine  wheel  W2,  and  as  this  wheel  revolves 


THE  GENERAL  NUMBER.  37 

with  velocity  s  =  400,  the  relative  velocity,  that  is,  the  velocity 
with  which  the  steam  enters  the  bucket  passages  of  wheel  W2,  is, 

85  =  84—8 

'  =(955  +  550/) -400 
=  555  +  550/  ft.  per  sec; 

and  is  represented  by  vector  OS5  in  Fig.  18. 
The  direction  of  this  steam  jet  is  given  by 

550 
tan  ^5  =  H55  =  0-990,     as    ^5  =  44.8  deg. 

Therefore,  the  entrance  edge  of  the  buckets  of  the  second 
wheel  W2  must  be  shaped  under  angle  (95=44.8  deg. 
The  absolute  value  of  the  entrance  velocity  is 


S5  =  V5552+5502  =  780  ft.  per  sec. 

In  traversing  the  bucket  passages,  the  velocity  drops  from 
the  entrance  value  85,  to  the  exit  value, 

S6  =  S5  (1  -  Ay)  =  780  X  0.88  =  690  ft.  per  sec. 

Since  the  exit  angles  of  the  buckets  of  wheel  W2  has  been 
chosen  as  45  cleg.,  and  the  exit  is  in  backward  direction,  6^  = 
180—45=135  deg.,  the  steam  jet  velocity  at  the  exit  of  the 
bucket  passages  of  the  last  wheel  is  given  by  the  general  number 

'56  =  86  (cos  de  +/  sin  de) 

=  690  (cos  135  deg.  +/  sin  135  deg.) 

=  -487+487/ ft.  per  sec, 

and  represented  by  vector  086  in  Fig.  18. 

Since  s  =  400  is  the  wheel  velocity,  the  velocity  of  the 
steam  after  leaving  the  last  wheel  W2,  that  is,  the  ''lost" 
or  ''  rejected  "  velocity,  is 

S7=Sq-\-S 

=  (-487+487/) +400 
=  -87  +  487/ ft.  per  sec, 

and  is  represented  by  vector  O87  in  Fig.  18. 


38 


ENGINE ERl  V G  M  A  THEM  A  TICS. 


The  direction  of  the  exhaust  steam  is  given  by, 

487 
tan^7= -- o-= -5.0,     as    ^7  =  180-80  =  100  deg., 

and  the  absolute  velocity  is, 


S7  =  \/872  +  4872  =  495  ft.  per  sec. 

Multiplication  of  General  Numbers. 

24.    If    A  =  ai+ya2   and    B  =  hi+jb2,   are    two    general,    or 
plane  numbers,  their  product  is  given  by  multiplication,  thus: 

AB  =  (ai+ja2)  (6 1+J62) 

=  «!&!  +yai62  +ya2&i  +j^a2b2, 
and  since  p=  —1, 

AB=  (aibi  — 02^2)  +y(ai&2+  a2&i), 

and  the  product  can  also  be  represented  in  the  plane,  by  a  point, 

C-=Ci+]'C2, 

ci  =aihi  — a2&2, 


where, 
and 


C2  =  aih2  H-aofei. 

For  instance,  .4=2+/  multiplied  by  5  =  1+1.5/  gives 

ci  =2X1 -1X1.5  =  0.5, 
C2  =  2Xl.5  +  lXl  =  4; 
hence, 

(7  =  0.5  +  4/, 

as  shown  in  Fig.  19. 

25.  The  geometrical  relation  between  the  factors  A  and  B 
and  the  product  C  is  better  shown  by  using  the  polar  expression; 
hence,  substituting. 


a\=a  cos  a 
a?  =  a  sin  a 


{vhich  gives 


,     hi=b  cos  f:l 
^"'^     62  =  6  sin  ^/' 


tan  a 


^2 


ai 


and 


tan^^  = 


h^Vb^Tb^ 

&2 


61 


THE  GENERAL  NUMBER. 


39 


the  quantities  may  be  written  thus  : 

A  =  a(eos  o:+/sin  a); 
J?  =  6(cos/?+ysin/?), 
and  then, 

C  =  AB  =  ab (cos  a+j'sin  «)(cos  /?+  j  sin  /?) 
=  ab  \  (cos  a  cos  /?  —sin  a  sin  /?)  +y(cos  a  sin  ^  +sin  a  cos  /3)| 
=  a6  {cos  («+/?)+/ sin  («+/?)}: 


Fig.  19.     ^lultiplication  of  Vectors. 

that  is,  two  general  numbers  are  multiplied  by  multiplying  their 
absolute  values  or  vectors,  a  and  6,  and  adding  their  phase  angles 
a  and  /?. 

Thus,  to  multiply  the  vector  quantity,  A  =  ai+ja2  =  a  {cos 
a  +y  sin  (i) by  5  =  6i  +  ^62  =  b  (cos  /?  +  /  sin  /?)  the  vector  0.4  in  Fig. 
19,  which  represents  the  general  number  A,  is  increased  by  the 
factor  h  =  Voi^  +  M,  and  rotated  by  the  angle  /?,  which  is  given 

bv  tan  B  =  T-- 

Thus,  a  complex  multiplier  B  turns  the  direction  of  the 
multiplicand  A,  by  the  phase  angle  of  the  multiplier  B,  and 
multiplies  the  absolute  value  or  vector  of  A^  by  the  absolute 
value  of  B  as  factor. 


40  ENGINEERING  MATHEMATICS. 

The  multiplier  B  is  occasionally  called  an  operator,  as  it 
carries  out  the  operation  of  rotating  the  direction  and  changing 
the  length  of  the  multiphcand. 

26.  In  multiplication,  division  and  other  algebraic  opera- 
tions with  the  representations  of  physical  quantities  (as  alter- 
nating currents,  voltages,  impedances,  etc.)  by  mathematical 
symbols,  whether  ordinary  numbers  or  general  numbers,  it 
is  necessary  to  consider  whether  the  result  of  the  algebraic 
operation,  for  instance,  the  product  of  two  factors,  has  a 
physical  meaning,  and  if  it  has  a  physical  meaning,  whether 
this  meaning  is  such  that  the  product  can  be  represented  in 
the  same  diagram  as  the  factors. 

For  instance,  3X4  =  12;  but  3  horses  X  4  horses  does  not 
give  12  horses,  nor  12  horses^,  but  is  physically  meaningless. 
However,  3  ft.  X4  ft.  =  12  sq.ft.     Thus,  if  the  numbers  represent 

"       ^ — I — I    0  (D — I — I — I — I — I — I — I — 0111        '■ 
0  A    B  C 

Fig.  20. 

horses,  multiplication  has  no  physical  meaning.  If  they  repre- 
sent feet,  the  product  of  multiplication  has  a  physical  meaning, 
but  a  meaning  which  differs  from  that  of  the  factors.  Thus, 
if  on  the  line  in  Fig.  20,  0A  =  3  feet,  0-S  =  4  feet,  the  product, 
12  square  feet,  while  it  has  a  physical  meaning,  cannot  be 
represented  any  more  by  a  point  on  the  same  line;  it  is  not 
the  point  0C  =  12,  because,  if  we  expressed  the  distances  OA 
and  OB  in  inches,  36  and  48  inches  respectively,  the  product 
would  be  36X48  =  1728  sq.in.,  while  the  distance  OC  would  be 
144  inches. 

27.  In  all  mathematical  operations  with  physical  quantities 
it  therefore  is  necessary  to  consider  at  every  step  of  the  mathe- 
matical operation,  whether  it  still  has  a  physical  meaning, 
and,  if  graphical  representation  is  resorted  to,  whether  the 
nature  of  the  physical  meaning  is  such  as  to  allow  graphical 
representation  in  the  same  diagram,  or  not. 

An  instance  of  this  general  limitation  of  the  appUcation  of 
mathematics  to  physical  quantities  occurs  in  the  representation 
of  alternating  current  phenomena  by  general  numbers,  or 
complex  quantities. 


THE  GENERAL  NUMBER. 


41 


An  alternating  current  can  be  represented  by  a  vector  01 
in  a  polar  diagram,  Fig.  21,  in  which  one  complete  revolution 
or  360  deg.  represents  the  time  of  one  complete  period  of  the 
alternating  current.  This  vector  01  can  be  represented  by  a 
general  number, 

where  ii  is  the  horizontal,  12  the  vertical  component  of  the 
current  vector  01. 


Fig.  21.     Current,  E.M.F.  and  Impedance  Vector  Diagram. 

In  the  same  manner  an  alternating  E.M.F.  of  the  same  fre- 
quency can  be  represented  by  a  vector  OE  in  the  same  Fig.  21, 
and  denoted  by  a  general  number, 

E  =  ei+je2. 

An  impedance  can  be  represented  by  a  general  number, 

Z  =  r-\-jx, 

where  r  is  the  resistance  and  x  the  reactance. 

If  now  we  have  two  impedances,  OZ^  and  OZ2,  Zi=ri-^]Xi 
and  Z2  =  r2-\-jx2,  their  product  Zi  Z2  can  be  formed  mathemat- 
ically, but  it  has  no  physical  meaning. 


42  ENGINEERING  MATHEMATICS. 

If  we  have  a  current  and  a  voltage,  I  =  i\-\-  ji2  and  E  =  e\-\-  je2, 
the  product  of  current  and  voltage  is  the  power  P  of  the  alter- 
nating circuit. 

The  product  of  the  two  general  numbers  /  and  E  can  be 
formed  mathematically,  IE,  and  would  represent  a  point  C 
in  the  vector  plane  Fig.  21.  This  point  C,  however,  and  the 
mathematical  expression  IE,  which  represents  it,  does  not  give 
the  power  P  of  the  alternating  circuit,  since  the  power  P  is  not 
of  the  same  frc^qucncy  as  /  and  E,  and  therefore  cannot  be 
represented  in  the  same  polar  diagram  Fig.  21,  which  represents 
I  and  E. 

If  we  have  a  current  /  and  an  impedance  Z,  in  Fig.  21; 
I  =  ii+ji2aJid  Z  =  r+jx,  their  product  is  a  voltage,  and  as  the 
voltage  is  of  the  same  frequency  as  the  current,  it  can  be  repre- 
sented in  the  same  polar  diagi'am.  Fig.  21,  and  thus  is  given  by 
the  mathematical  product  of  [  and  Z, 

E  =  IZ=(i,+ji2){r^-jx), 

=  (iir  —iiX  )  +j(i2r +iix). 

28.  Commonly,  in  the  denotation  of  graphical  diagrams  by 
general  numbers,  as  the  polar  diagram  of  alternating  currents, 
those  quantities,  which  are  vectors  in  the  polar  diagram,  as  the 
current,  voltage,  etc.,  are  represented  by  dotted  capitals:  E,  I, 
while  those  general  numbers,  as  the  impedance,  admittance,  etc. , 
which  appear  as  operators,  that  is,  as  multipliers  of  one  vector, 
for  instance  the  current,  to  get  anather  vector,  the  voltage,  are 
represented  algebraically  by  capitals  without  dot:  Z  =  r-ijx^ 
impedance,  etc. 

This  limitation  of  calculation  with  the  mathematical  repre- 
sentation of  physical  quantities  must  constantly  be  kept  in 
mind  in  all  theoretical  investigations. 

Division  of  General  Numbers. 

29.  The  division  of  two  general  numbers,  A=ai-\rja2  and 
B  =  bi+jb2,  gives, 

A     ai+jao 
'~  B~b,+jb2 
This  fraction  contains  the  quadrature  number  in  the  numer- 
ator as  well  as  in  the  denominator.     The  quadrature  number 


THE  GENERAL  NUMBER.  43 

can  be  eliminated  from  the  denominator  by  multiplying  numer- 
ator and  denominator  by  the  conjugate  quantity  of  the  denom- 
inator, 61-/62,  which  gives: 

^     (ai  -\-ja2)(b\  —jbo)     (a\hi  +0262)  +j(a2bi  —aibj) 
'^{bi+jb2){b,-jb2)~  b,^+b2' 

a]bi  +a2b2      .  ^261  —0162 

~  ~^  2  A  h'2        '  i 


for  instance, 


bi^  +  b2 


A^Q  +  2.5j 
'~B'  3  +  4/ 

_(6  +  2.5/)(3-^/) 
(3+4/)(3-4/) 
28-16.5/ 
25 
=  1.12-0.00/. 

If  desired,  the  quadrature  number  may  be  eliminated  from 
the  numerator  and  left  in  the  denominator  by  multiplying  with 
the  conjugate  number  of  the  numerator,  thus: 

A     ai+ja2 
•  ~^~6i+/62 

{ax+ja2)(a\  -jaj) 

~  {bi+jb2){ai-ja2) 

ai^+a2^ 


for  instance. 


{aibi  -\ra2h2)  +j(,aib2  -0261) ' 


A     0  +  2.5/ 
•~B~  3+4/ 

*      (0  +  2.5/)  (6 -2.5,?') 
~  (3 +  4/) (0-2.5/) 
42.25 


28  +  10.5/ 


30.  Just  as  in  multiplication,  the  polar  representation  of 
the  general  number  in  division  is  more  perspicuous  than  any 
other. 


44  ENGINEERING  MATHEMATICS. 

Let  A=a(cos  a+/sin  a)  be  divided  by  B  =  b{cos  ,d+j  sm  ^), 
thus: 

A     a{cos  a  +  j  sin  a) 

a(cos  a+j  sin  a) (cos  /?  —  j  sin  /?) 
"6(008  /? +/  sin  /?)  (cos  /?  —  j  sin  ^) 

a{  (cos  a  cos  /?  +  sin  a  sin  /?)  +j(sin  a  cos  /?— cos  a:  sin  /?)  1 
"^  6(cos^/3  +  sin2/3) 

=  t|cos  («— /?)+ysin  (a— /?)|. 

That  is,  general  numbers  A  and  B  are  divided  by  dividing 
their  vectors  or  absolute  values,  a  and  b,  and  subtracting  their 
phases  or  angles  a  and  /?. 

Involution  and  Evolution  of  General  Numbers. 

31.  Since  involution  is  multiple  multiplication,  and  evolu- 
tion is  involution  vdth  fractional  exponents,  both  can  be  resolved 
into  simple  expressions  by  using  the  polar  form  of  the  general 
number. 

A=ai  +ya2  =  a(cos  a  +j  sin  a), 
then 

(7  =  A"  =  a'*(cos  na:+/sin  na). 

For  instance,  if 

A  =  3 +4/ =  5 (cos  53  deg. +/sin  53  deg.); 
then, 

C'  =  A4  =  5*(cos  4  X53  deg.  +j  sin  4  X53  deg.) 
=  625 (cos  212  deg.  +/  sin  212  deg.) 
=  625(  -cos  32  deg.  -/  sin  32  deg.) 
=  625( -0.848 -0.530/) 
=  -529-331  J. 

If,  A  =ai  +ja2  =  a  (cos  a  +J  sin  a),  then 

C=  \/A  =  A""  =  a"^( cos  -  +  7  sin  - ) 

nr-(       «  ,  •  •     «\ 
=  val  cos  —  4-7  sm  —  I. 

\       V.     ^        nl 


THE  GENERAL  NUMBER.  45 

32.  If,  in  the  polar  expression  of  A,  we  increase  the  phase 
angle  a  by  27r,  or  by  any  multiple  of  2;r :  2g;r,  where  q  is  any 
integer  number,  we  get  the  same  value  of  A,  thus: 

A  =  a{cos{a+2q7T)  +/ sin(a+2g7r)|, 

since  the  cosine  and  sine  repeat  after  every  360  deg,  or  2-. 
The  nth  root,  however,  is  different: 

ry      «/-T      »^f        oc+2q7:      .   .     a+2q7:\ 

C  =  vA  =  va(  cos ~  +1  sm ~  I. 

\  n         ■'  n     I 

We  hereby  get  n  different  values  of  C,  for  <7  =  0,  1,  2.  ,  .n— 1; 
q  =  n  gives  again  the  same  as  g  =  0.     Since  it  gives 

a  +  2n-z     a     ^ 

=-^-2^l\ 

n         n 

that  is,  an  increase  of  the  phase  angle  by  360  deg.,  which  leaves 
cosine  and  sine  unchanged. 

Thus,  the  nth  root  of  any  general  number  has  n  different 
values,  and  these  values  have  the  same  vector  or  absolute 

term  v^a,  but  differ  from  each  other  by  the  phase  angle  —  and 

its  multiples. 

For  instance,  let  4= -529 -331/  =  625  (cos  212  deg.  4- 
/  sin  212  deg.)  then, 

n      M-T      *f7r^l       212 +  360^      .  .     212+3605\ 
C=  a/A=  i625(cos r ^  +  jsm j ^1 

=  5(cos53+/sin53)  =3  +  4/ 

=  5(cos  143  +/  sin  143)  =  5(  -cos  37  +  /  sin  37)  =  -4  +3/ 
=  5(cos  233  +  / sin  233)  =5( -cos  53-/ sin  53)  =  -3-4/ 
=  5(cos  323  +  /  sin  323)  =  5(cos  37  -/  sin  37)     =4-3/ 
=  5(cos  413  +/  sin  413)  =  5(cos  53  +  /  sin  53)     =3  +  4/ 

The  n  roots  of  a  general  number  A  =a(cos  a  +/  sin  a)  differ 

*  2tz 
from  each  other  by  the  phase  angles  — ,  or  1/nth  of  360  deg., 

and  since  they  have  the  same  absolute  value  v^a,  it  follows,  that 
they  are  represented  by  n  equidistant  points  of  a  circle  with 
radius  \^,  as  shown  in  Fig.  22,  for  77=4,  and  in  Fig.  23  for 


46 


ENGINEERING  MA  THEM  A  TICS. 


n  =  9.  Such  a  system  of  n  equal  vectors,  tliffering  in  phase  from 
each  other  by  1/nth  of  360  cleg.,  is  called  a.  polyphase  system,  or 
an  n-phase  system.  The  n  roots  of  the  general  number  thus 
give  an  n-phase  system. 

33.  For  instance,  \/l  =  ? 

If  A  =  a  (cos  a+jsin  a)  =  l.  this  means:  a=l,  «=0;   and 
hence, 

„/-  2q7:         .     2qz, 

V 1  =  cos 1-  7  sm , 

n      •"  n 


>,=3+4i 


P,=-4+3i 


Pi=4-3J 


Ps=-3-4J 


Fig.  22.     Roots  of  a  General  Number,  n  =  4. 
and  the  n  roots  of  the  unit  are 

5  =  0  <yi  =  l; 

360     .  .    360 
0  =  1  cos l-?sm— ; 

^360     .  .    ^     360 
0=2  cos2x — +7sm2x — ; 


q  =  n  —  l 
However, 


,       ^,360     .  .     ,       ,,360 

cos  (n  — 1) h?  sm  (n  — 1)  — . 


360 


360 


cos  q \-j  sm  q  —  =  I  cos 1- ?  sm 


360     .   .    360\9 


THE  GENERAL  NUMBER. 

hence,  the  n  roots  of  1  are, 

n/T     I       360     .  .    3G0\<' 
vl  =  ^cos  —^  -\-]  sin  —^\  , 


47 


n 


n 


where  q  may  be  any  integer  number. 

One  of  these  roots  is  real,  for  g'=0,  and  is  =  + 1. 

If  n  is  odd,  all  the    other    roots  are   general,  or  complex 
numbers. 


n   . 


If  n  is  an  even  number,  a  second  root,  for  g  =  ^,  is  also  real; 
cos  180 +y  sin  180= -1. 


Fig.  23.     Roots  of  a  General  Number,  n  =  g. 

If  n  is  divisible  by  4,  two  roots  are  quadrature  numbers,  and 

are    +/,    for  q=-7,  and  —j,  for  (Z=x- 

34.  Using  the  rectangular  coordinate  expression  of  the 
general  number,  A  ^ai  +ja2,  the  calculation  of  the  roots  becomes 
more    complicated.     For    instance,    given  ^(^4  =  ? 

Let  C='^.=Cl+jc2•, 

then,  squaring, 

A  =  (ci+yc2)2; 
hence, 

Ol  +ya2  =  (Cl^  ~C2^)  +2]'CiC2. 

Since,  if  tw^o  general  numbers  are  equal,  their  horizontal 
and  their  vertical  components  must  be  equal,  it  is: 

ai=c\^  —C2^    and      a2  =  2ciC2. 


48  ENGINEERING  MATHEMATICS. 

Squaring  both  equations  and  adding  them,  gives, 

Hence :  

and  since  ci^  — C2-  =  ai ', 


then,  ci^  =  \{VaJ+a^+ai), 


and  C2-  =  K'^W  +  a22-ai). 

Thus 


and 
and 


ci^y  l{\  ar  +  a2^  +  ai] 


which  is  a  rather  complicated  expression. 

35.  When  representing  physical  quantities  by  general 
numbers,  that  is,  complex  quantities,  at  the  end  of  the  calcula- 
tion the  final  result  usually  appears  also  as  a  general  number, 
or  as  a  complex  of  general  numbers,  and  then  has  to  be  reduced 
to  the  absolute  value  and  the  phase  angle  of  the  physical  quan- 
tity. This  is  most  conveniently  done  by  reducing  the  general 
numbers  to  their  polar  expressions.  For  instance,  if  the  result 
of  the  calculation  appears  in  the  form. 


■~  (di+yd2P(ei+ye2) 

by  substituting 

/ — n 7  ^2 

a  =  Vai^  +  a2^;     tana:  =  — . 

ai 


6  =  \/6i^  +  622;     tan/9  =  r^; 
and  so  on. 

a(cos  a  +/sin  q;)6^(cos  /9  +  ysin  .9)3\^(cos  ;'  +  ysin  /-)* 


R  = 


(i2(cos  o+jsin  3)'-e{cos  e+j sm  s) 
—^\cos{a+S!3  +  r/2  -2d  -  e)+j  sin  (a+3^+r/2  -2d  -e)\. 


THE  GENERAL  NUMBER.  49 

Therefore,  the  absohite  value  of  a  fractional  expression  is 
the  product  of  the  absolute  values  of  the  factors  of  the  numer- 
ator, divided  by  the  product  of  the  absolute  values  of  the 
factors  of  the  denominator. 

The  phase  angle  of  a  fractional  expression  is  the  sum  of 
the  phase  angles  of  the  factors  of  the  numerator,  minus  the  sum 
of  the  phase  angles  of  the  factors  of  the  denominator. 

For  instance. 


^   (3-4y)2(2+2y)-e/-2.5+6y 

5(4+3/)2\/2 
25(cos307+/sin307)22\/2"(cos45+ysin4o)-v/6!5(cosll4+ysinll4)* 

125  (cos  37+/ sin  37)  2  \/2 

=  0.4^^5 1  cos (2  X  307  +  45  + -^  - 2  X  37  j 

+  /sin  (2x307+45+^-2x37)  I 

=  0.4 ^6^ { cos  263  +  /  sin  263 } 

=-0.7461  -0.122-0.992/1  =  -0.091  -0.74/. 

36.  As  will  be  seen  in  Chapter  II: 

y?     u^     w* 

^"=i+^+l2-+^+jT+--- 

x^     r^     x^     i^ 

cosi-  =  l-^+j^-pr  +  p--+... 

T         or         T^ 
sin  X  =  X—'j:t-  +7^  — r;^  -\ .  .   . 

Herefrom  follows,  by  substituting,  x  =  d,  u  =  jd, 
cos  ^+/sin  d  =  e^\ 
and  the  polar  expression  of  the  complex  quantity, 

A  =a(cos  «  +/  sin  a), 
thus  can  also  be  written  in  the  form, 


50  ENGINEERING  MATHEMATICS. 

where  e  is  the  base  of  the  natural  logarithms, 

£  =  1+1+7^+^+^  +  .  .  .=2.71828.  .  . 

p       \6       |4 

Since  any  number  a  can  be  expressed  as  a  power  of  any 
other  number,  one  can  substitute, 

where  ao  =  \os,ta=T-^ ,  and  the   general    number  thus    can 

logio  ' 

also  be  written  in  the  form, 

that  is  the  general  number,  or  complex  quantity,  can  be  expressed 
in  the  forms, 

A  =ai  +ja2 
=  a(cos  a+j  sin  a) 

The  last  two,  or  exponential  forms,  are  rarely  used,  as  they 
are  less  convenient  for  algebraic  operations.  They  are  of 
importance,  however,  since  solutions  of  differential  equations 
frequently  appear  in  this  form,  and  then  are  reduced  to  the 
polar  or  the  rectangular  form. 

37.  For  instance,  the  differential  equation  of  the  distribu- 
tion of  alternating  current  in  a  flat  conductor,  or  of  alternating 
magnetic  flux  in  a  flat  sheet  of  iron,  has  the  form : 

and  is  integrated  by  y  =  Ae~ ^'^,  where, 


V=\/-2jc^  =  ±{l-j)c; 
hence, 

This  expression,  reduced  to  the  polar  form,  is 

i/  =  ^i£"'""(cos  ex  —j  sin  ex)  +A2£~"(cos  cx+j  sin  ex). 


THE  GENERAL  NUMBER.  51 

Logarithmation. 

38.  In  taking  the  logarithm  of  a  general  number,  the  ex- 
ponential expression  is  most  convenient,  thus : 

loge  (tti  +;a2)  =log£ a(cos  a+j  sin  a) 

=  log£  a+log££'" 
=  log£a+ya; 

or,  if  6  =  base  of  the  logarithm,  for  instance,  6  =  10,  it  is: 

logj,(ai  +ya2)  =logja£'''  =  logj,  o+ja  logj,  e; 

or,  if  6  unequal  10,  reduced  to  logio; 

1        /      .  •    \     logio  a      .   login  £ 
logio  b    •'    logio  b 


Note.  In  mathematics,  for  quadrature  unit  V  —  1  is  always 
chosen  the  symbol  i.  Since,  however,  in  engineering  the  symbol  i 
is  universally  used  to  represent  electric  current,  for  the  quatl- 
rature  unit  the  symbol  j  has  been  chosen,  as  the  letter  nearest 
in  appearance  to  i,  and  j  thus  is  always  used  in  engineering 
calculations  to  denote  the  quadrature  unit  V  — 1. 


CKA.PTER  II. 

POTENTIAL   SERIES   AND    EXPONENTIAL    FUNCTION. 

A.    GENERAL. 
39.  An  expression  such  as 

^'^r^ (1) 

represents  a  fraction;  that  is,  the  result  of  di\ision,  and  like 
any  fraction  it  can  be  calculated;  that  is,  the  fractional  form 
eliminated,  bydi\iding  the  numerator  by  the  denominator,  thus: 

l-x\l  =  l+x+x2+x^+.  .  . 
1-x 


+  x 

x 

-J2 

+  X2 

J2- 

-J-3 

+  X3. 

Hence,  the  fraction  (1)  can  also  Ije  expressed  in  the  form: 

y  =  ^—  =  l+x  +  x^  +  x^  + (2) 

This  is  an  infinite  series  of  successive  powers  of  x,  or  a  poten- 
tial series. 

In  the  same  manner,  by  di^^ding  through,    the  expression 

y'lhc <^> 

can  be  reduced  to  the  infinite  series, 

y  =  ,r^—  =  l-x  +  x^-r'+- (4) 

^     1+x 

52 


POTENTIAL  SERIES  AND  EXPONENTIAL  FUNCTION.     53 

The  infinite  series  (2)  or  (4)  is  another  form  of  representa- 
tion of  the  expression  (1)  or  (3),  just  as  the  periodic  decimal 
fraction  is  another  representation  of  the  common  fraction 
(for  instance  0.6363 =7/11). 

40.  As  the  series  contains  an  infinite  number  of  terms, 
in  calculating  numerical  values  from  such  a  series  perfect 
exactness  can  never  be  reached:  since  only  a  finite  number  of 
terms  are  calculated,  the  result  can  only  be  an  approximation. 
By  taking  a  sufficient  number  of  terms  of  the  series,  however, 
the  approximation  can  be  made  as  close  as  desired;  that  is, 
numerical  values  may  be  calculated  as  exactly  as  necessary, 
so  that  for  engineering  purposes  the  infinite  series  (2)  or  (4) 
gives  just  as  exact  numerical  values  as  calculation  by  a  finite 
expression  (1)  or  (2),  provided  a  suflficient  number  of  terms 
are  used.  In  most  engineering  calculations,  an  exactness  of 
0.1  per  cent  is  sufficient;  rarely  is  an  exactness  of  0.01  per  cent 
or  even  greater  required,  as  the  unavoidable  variations  in  the 
nature  of  the  materials  used  in  engineering  structures,  and  the 
accuracy  of  the  measuring  instruments  impose  a  limit  on  the 
exactness  of  the  result. 

For  the  value  x  =  0.5,  the  expression  (1)  gives  y  =  z — jr-z  =  2; 

while  its  representation  by  the  series  (2)  gives 

y  =  1+0.5+0.25+0.125+0.0625+0.03125  +  .  .  .  (5) 

and  the  successive  approximations  of  the  numerical  ^'alues  of 
y  then  arc : 

using  one  term:  J/=l  =1;  error:  —1 

"      two  terms:  ?/=  1  +  0.5  =1.5;  "  —0.5 

"      three  terms:  i/=  1  +  0.5  + 0.25  =1.75-  "  -0.25 

"      four  terms:  ?/=  1  +  0.5  +  0.25  +  0.125  =1.875;        "  -0.125 

"      five  terms:  ?/=  1  +  0.5+0.25+0.125  +  0.0625=  1.9375       "  -0.0625 

It  is  seen  that  the  successive  approximations  come  closer  and 
closer  to  the  correct  value,  y  =  2,  but  in  this  case  always  remain 
below  it;  that  is,  the  series  (2)  approaches  its  limit  from  below, 
as  shown  in  Fig.  24,  in  which  the  successive  approximations 
are  marked  by  crosses. 

For  the  value  x  =  0.5,  the  approach  of  the  successive 
approximations  to  the  limit  is  rather  slow,  and  to  get  an  accuracy 
of  0.1  per  cent,  that  is,  bring  the  error  down  to  less  than  0.002, 
requires  a  considerable  number  of  terms. 


54  ENGINEERING  MATHEMATICS. 

For  x  =  0.\  the  series  (2)  is 

?/  =  l +0.1 +0.01 +0.001 +0.0001+ (6) 

and  the  successive  approximations  thus  are 

l:y=l;  2.y=l.l;  3:i/=l.ll;   4:t/=l.lll;    5:t/=l.llll; 
and  as,  by  (1),  the  final  or  limiting  value  is 

1  10 

^=r=oT=-9=^-^^^^--- 


+  3 

2 

*1  *    l-a? 


Fig.  24.     Convergent  Series  with  One-sided  Approach. 

the  fourth  approximation  already  brings  the  error  well  below 
0.1  per  cent,  and  sufficient  accuracy  thus  is  reached  for  most 
engineering  purposes  by  using  four  terms  of  the  series. 
41.    The  expression  (3)  gives,  for  x  =  0.5,  the  value, 

Represented  by  series  (4),  it  gives 

t/  =  1  -0.5  +  0.25  -0.125  +  0.0625  -0.03125  +  - (7) 

the  successive  approximations  are; 

1st:  T/=l  =1;  error:  +0.333... 

2d:  y=l-0.5  =0.5;  "  -0.1666... 

3d:  ?/=  1-0.5+0.25  =0.75;         "  +0.0833... 

4th:  7/-  1-0.5+0.25-0.125  =0.625;       "  -0.04166... 

5th:  j/=  1-0.5  +  0.25-0.125  +  0.0625  =  0.6875;     "  +0.020833... 

As  seen,  the  successive  approximations  of  this  scries  come 
closer  and  closer  to  the  correct  value  ?/  =  0.6666  .  .  .  ,  but  in  this 
case  are  alternately  above  and  below  the  correct  or  limiting 
value,  that  is,  the  series  (4)  approaches  its  limit  from  both  sides, 
as  shown  in  Fig.  25,  while  the  series  (2)  approached  the  limit 
from  below,  and  still  other  series  may  approach  their  hmit 
from  above. 


POTENTIAL  SERIES  AND  EXPONENTIAL  FUNCTION.     55 

With  such  alternating  approach  of  the  series  to  the  limit, 
as  exhibited  by  series  (4),  the  limiting  or  final  value  is  between 
any  two  successive  approximations,  that  is,  the  error  of  any 
approximation  is  less  than  the  difference  between  this  and  the 
next  following  approximation. 

Such  a  series  thus  is  preferable  in  engineering,  as  it  gives 
information  on  the  maximum  possible  error,  while  the  series 
with  one-sided  approach  does  not  do  this  without  special  in- 
vestigation, as  the  error  is  greater  than  the  difference  between 
successive  approximations. 

42.  Substituting  x  =  2  into  the  expressions  (1)  and  (2), 
equation  (1)  gives 


*1 


3 
+  5 

±_ 


*  1 

*    1+x 

Fig.  25.     Convergent  Series  with  Alternating  Approach. 

while  the  infinite  series  (2)  gives 

y  =  l+2-l-4+8  +  lG+32  +  .  ..; 
and  the  successive  approximations  of  the  latter  thus  are 

1;  3;  7;  15;  3i;  G3.  .  .; 
that  is,  the  successive  approximations  do  not  approach  closer 
and  closer  to  a  final  value,  but,  on  the  contrary,  get  further  and 
further  away  from  each  other,  and  give  entirely  wrong  results. 
They  give  increasing  positive  values,  which  apparently  approach 
00  for  the  entire  series,  while  the  correct  value  of  the  expression, 
by  (1),  is  y=  -1. 

Therefore,  for  x  =  2,  the  series  (2)  gives  unreasonable  results, 
and  thus  cannot  be  used  for  calculating  numerical  values. 

The  same  is  the  case  with  the  representation   (4)  of    the 
expression  (3)  for  x  =  2.     The  expression  (3)  gives 

V  =  r^,  =  0.3333.  .  .  ; 


56  ENGINEERING  MATHEMATICS. 

while  the  infinite  series  (4)  gives 

?/  =  1-2+4-8  +  16-32+  -.  .., 

and  the  successive  approximations  of  the  latter  thus  are 

1;     -1;     +3;     -5;     +11;     -21;  .  .  .: 

hence,  while  the  successive  values  still  are  alternately  above 
antl  below  the  correct  or  limiting  value,  they  do  not  approach 
it  with  increasing  closeness,  but  more  and  more  diverge  there- 
from. 

Such  a  series,  in  which  the  values  derived  by  the  calcula- 
tion of  more  and  more  terms  do  not  approach  a  final  value 
closer  and  closer,  is  called  divergent,  while  a  series  is  called 
convergent  if  the  successive  approximations  approach  a  final 
value  with  increasing  closeness. 

43.  While  a  finite  expression,  as  (1)  or  (3),  holds  good  for 
all  values  of  x,  and  numerical  values  of  it  can  be  calculated 
whatever  may  be  the  value  of  the  independent  variable  x,  an 
infinite  series,  as  (2)  and  (4),  frequently  does  not  give  a  finite 
result  for  every  value  of  x,  but  only  for  values  within  a  certain 
range.  For  instance,  in  the  above  series,  for  —  1  <x<  +  l, 
the  series  is  convergent;  while  for  values  of  x  outside  of  this 
range  the  series  is  divergent  and  thus  useless. 

When  representing  an  expression  by  an  infinite  series, 
it  thus  is  necessary  to  determine  that  the  series  is  convergent; 
that  is,  approaches  with  increasing  number  of  terms  a  finite 
limiting  value,  otherwise  the  series  cannot  be  used.  Where 
the  series  is  convergent  within  a  certain  range  of  x,  diver- 
gent outside  of  this  range,  it  can  be  used  only  in  the  range  oj 
convergency,  but  outside  of  this  range  it  cannot  be  used  for 
deriving  numerical  values,  but  some  other  form  of  representa- 
tion has  to  be  found  which  is  convergent. 

This  can  frequently  be  done,  and  the  expression  thus  repre- 
sented by  one  series  in  one  range  and   by  another  series    in 

another  range.     For  instance,  the  expression  (1),  ?/ =  7-77-7  by 

substituting,  x  =  -,  can  be  written  in  the  form 

1  u 


1+- 

u 


POTENTIAL  SERIES  AND  EXPONENTIAL  FUNCTION.     57 

and  then  developed  into  a  series  by  dividing  the  numerator 
by  the  denominator,  which  gives 

y  =  u  —  u^+u^~u'^  +  .  .  . ; 

or,  resubstituting  x, 

1111 

^=x-F^+F3--4  +  ---,  ....    (8) 

which  is  convergent  for  x  =  2,  and  for  x  =  2  it  gives 

?/  =  0.5 -0.25  +  0.125 -0.0625 +  .  .  .       (9) 
With  the  successive  approximations : 

0.5;    0.25;     0.375;     0.3125..., 
which  approach  the  final  Umiting  value, 

7/  =  0.333.  .. 

44.  An  infinite  series  can  be  used  only  if  it  is  convergent. 
Mathemetical  methods  exist  for  determining  whether  a  scries 
is  convergent  or  not.  For  engineering  purposes,  however, 
these  methods  usually  are  unnecessary;  for  practical  use  it 
is  not  sufficient  that  a  series  be  convergent,  but  it  must  con- 
verge so  rapidly — that  is,  the  successive  terms  of  the  series 
must  decrease  at  such  a  gi'cat  rate — that  accurate  numerical 
results  are  derived  by  the  calculation  of  only  a  very  few  terms; 
two  or  three,  or  perhaps  three  or  four.  This,  for  instanco, 
is  the  case  with  the  series  (2)  and  (4)  for  j:  =  0.1  or  less.  For 
x  =  0.5,  the  series  (2)  and  (4)  are  still  convergent,  as  seen  in 
(5)  and  (7),  but  are  useless  for  most  engineering  purposes,  as 
the  successive  terms  decrease  so  slowly  that  a  large  number 
of  terms  have  to  be  calculated  to  get  accurate  results,  and  for 
such  lengthy  calculations  there  is  no  time  in  engineering  work. 
If,  however,  the  successive  terms  of  a  series  decrease  at  such 
a  rapid  rate  that  all  but  the  first  few  terms  can  be  neglected, 
the  series  is  certain  to  be  convergent. 

In  a  series  therefore,  in  which  there  is  a  question  whether 
it  is  convergent  or  divergent,  as  for  instance  the  series 

,11111 

2/  =  l+2 +3 +4 +5 +e  ^'  '  ■  (^^^^^^g^^*)' 


58  ENGINEERING  MATHEMATICS. 

or 

1      1      1      1      1      1 

?/  =  !--+-  -^+^-^+.  .  .  (convergent), 

the  matter  of  convergency  is  of  little  importance  for  engineer- 
ing calculation,  as  the  series  is  useless  in  any  case;  that  is,  does 
not  give  accurate  numerical  results  with  a  reasonably  moderate 
amount  of  calculation. 

A  series,  to  be  usable  for  engineering  work,  must  have 
the  successive  terms  decreasing  at  a  very  rapid  rate,  and  if 
this  is  the  case,  the  series  is  convergent,  and  the  mathematical 
investigations  of  convergency  thus  usually  becomes  unnecessary 
in  engineering  work. 

45.  It  would  rarely  be  advantageous  to  develop  such  simple 
expressions  as  (1)  and  (3)  into  infinite  series,  such  as  (2)  and 
(4),  since  the  calculation  of  numerical  values  from  (1)  and  (3) 
is  simpler  than  from  the  series  (2)  and  (4),  even  though  very 
few  terms  of  the  series  need  to  be  used. 

The  use  of  the  series  (2)  or  (4)  instead  of  the  expressions 
(1)  and  (3)  therefore  is  advantageous  only  if  these  series  con- 
verge so  rapidly  that  only  the  first  two  terms  are  required 
for  numerical  calculation,  and  the  third  term  is  negligible; 
that  is,  for  verv  small  values  of  x.  Thus,  for  x  =  0.01,  accord- 
ing to  (2), 

2/  =  l +0.01 +0.0001 +.  .  .  =  1+0.01, 

as  the  next  term,  0.0001,  is  already  less  than  0.01  per  cent  of 
the  value  of  the  total  expression. 

For  very  small  values  of  x,  therefore,  by  (1)  and  (2), 

^=j--  =  l+-^, (^^) 

and  by  (3)  and  (4), 

y-Yv.-'-''' (") 

ana  tnese  expressions  (10)  and  (11)  are  useful  and  very  com- 
monly used  in  engineering  calculation  for  simplifying  work. 
For  instance,  if  1  plus  or  minus  a  very  small  quantity  appears 
as  factor  in  the  denominator  of  an  expression,  it  can  be  replaced 
b)'  1  minus  or  plus  the  same  small  quantity  as  factor  in  the 
numerator  of  the  expression,  and  inversely. 


POTENTIAL  SERIES  AND  EXPONENTIAL  FUNCTION.     59 

For  example,  if  a  direct-current  receiving  circuit,  of  resist- 
ance r,  is  fed  by  a  supply  voltage  eo  over  a  line  of  low- 
resistance  ro,  what  is  the  voltage  e  at  the  receiving  circuit? 

The  total  resistance  is  r  +  ro;    hence,  the  current,  i= — —. 

r  +  ro' 
and  the  voltage  at  the  receiving  circuit  is 

r 

e  =  n  =  eo— — • (12) 

r  +  ro 

If  now  ro  is  small  compared  with  r,  it  is 

e  =  eo—— =60(1-7} (13) 

r 


As  the  next  term  of  the  series  would  be  (  — I  ,  the  error 


made  by  the  simpler  expression  (13)  is  less  than  (— j  .  Thus, 
if  ro  is  3  per  cent  of  r,  which  is  a  fair  average  in  interior  light- 
ing circuits,  (  — )  =0.03^  =  0.0009,  or  less  than   0.1  per  cent; 

hence,  is  usually  negUgible. 

46.  If  an  expression  in  its  finite  form  is  more  complicated 
and  thereby  less  convenient  for  numerical  calculation,  as  for 
instance  if  it  contains  roots,  development  into  an  infinite  series 
frequently  simplifies  the  calculation. 

Very  convenient  for  development  into  an  infinite  series 
of  powers  or  roots,  is  the  binomial  theorem, 


{l±u)''  =  l±nu-\ r^ —  u^±~ j^ ir  +  . 

where 

|m  =  lX2x3X.  .  .Xm. 


(14) 


Thus,    for   instance,    in    an    alternating-current    circuit    of 
resistance  r,  reactance  x,  and  supply  voltage  e,  the  current  is. 


60  ENGINEERING  MATHEMATICS. 

If  this  circuit  is  practically  non-inductive,  as  an  incandescent 
lighting  circuit;  that  is,  if  x  is  small  compared  with  r,  (15) 
can  be  written  in  the  form, 

and  the  square  root  can  be  developed  by  the  binomial  (14),  thus, 
21  =  (-)  ;  n=  — -,  and  gives 

In  this  series  (17),  if  x  =  0.lr  or  less;  that  is,  the  reactance 
is  not  more  than  10  per  cent  of  the  resistance,  the  third  term, 

^  ( - )  ,  is  less  than  0.01   per  cent;    hence,  negligible,  and  the 

series  is  approximated  with  sufficient  exactness  by  the  firet 
two  terms. 


and  equation  (16)  of  the  current  then  gives 


14(7).     •     •     •     •     (18) 


b-m (-) 


r 


This  expression  is  simpler  for  numerical  calculations  than 
the  expression  (15),  as  it  contains  no  square  root. 

47.  Development  into  a  series  may  become  necessary,  if 
further  operations  have  to  be  carried  out  with  an  expression 
for  which  the  expression  is  not  suited,  or  at  least  not  well  suited. 
This  is  often  the  case  where  the  expression  has  to  be  integrated, 
since  very  few  expressions  can  be  integrated. 

Expressions  under  an  integral  sign  therefore  very  commonly 
have  to  be  developed  into  an  infinite  series  to  carry  out  the 
integration. 


POTENTIAL  SERIES  AND  EXPONENTIAL  FUNCTION.     61 


EXAMPLE    1. 

Of  the  equilateral  hyperbola  (Fig.  26), 


xy  =  a^, 


(20) 


the  length  L  of  the  arc  loctween    3-1=  2a  and  2*2  =  10a  is  to  be 
calculated. 

An  element  dl  of  the  arc  is  the  hypothenuse  of  a  right  triangle 
with  dx  and  dy  as  cathetes.     It,  therefore,  is. 


dl=Vdx^+dy^ 


-Ml^^'^^' (^^> 


\ 

\ 

\ 

\ 

\ 

\ 

\ 

K 

d^ 

\ 

f 

^ 

^^ 

xy= 

:a" 

— 

— 

, 

X, 

X 

v., 

Fig.  26,     Equilateral  Hyperbola. 


and  from  (20), 


y 


a^ 


,    dy      0,^ 

and     -r=  — t^. 


X  dx 

Substituting  (22)  in  (21)  gives, 


hence,  the  length  L  of  the  arc,  from  xi  to  X2  is. 


'^=i?=£i^(jh- 


(22) 


(23) 


62  ENGINEERING  MATHEMATICS. 


Substituting  -  =  i';  that  is,  dx  =  adv,  also  substituting 

?;i=— =2     and     ?;2  =  — =10, 
a  a 


gives 

r^-  I    r 

dv. 


^«X^R 


The  expression  under  the  integral  is  inconvenient  for  integra- 
tion; it  is  preferably  developed  into  an  infinite  series,  by  the 
binomial  theorem  (14). 

Write  w  =  —r     and     n  =  :r,  then 


r" 


'?' 


]__      J. 1_    _J 5_ 

and 

[.11  1 

=  av{  1- , .   .+ 


2X3XV*      7X8XV«     llXl6Xt'- 

1 


3X128X^16 


-  + 


L        xi/i      lA     1/1     i\ 


and  substituting  the  numerical  values, 
L  =  a\  (10-2) +^(0.125-0.001) 

--^(0.0078-0) +^(0.0001  -0)1 

=  a{8  +  0.0207  -  0.0001 }  =  8.0206a. 

As  seen,  in  this  series,  only  the  first  two  terms  are  appreciable 
in  value,  the  third  term  less  than  0.01  per  cent  of  the  total, 
and  hence  negligible,  therefore  the  series  converges  very 
rapidly,  and  numerical  values  can  easily  be  calculated  by  it. 


POTENTIAL  SERIES  AND  EXPONENTIAL  FUNCTION.     63 

For  xi<2  a]  that  is,  Vi  <2,  the  series  converges  less  rapidly, 
and  becomes  divergent  for  Xi<a;  or,  ri<l.  Thus  this  series 
(17)  is  convergent  for  v>  1,  but  near  this  limit  of  convergency 
it  is  of  no  use  for  engineering  calculation,  as  it  does  not  converge 
with  sufficient  rapidity,  and  it  becomes  suitable  for  engineering 
calculation  only  when  vi  approaches  2. 

EXAMPLE    2. 

48.  log  1=0,  and,  therefore  log  (1+x)  is  a  small  quantity 
if  X  is  small,  log  (l+x)  shall  therefore  be  developed  in  such 
a  series  of  powers  of  x,  which  permits  its  rapid  calculation 
without  using  logarithm  tables. 

It  is 

log  u=J-; 
then,  substituting  (l+x)  for  u  gives, 

log(l+x)=Jp_^ (24) 


From  equation  (4) 
1 


=  l  —  x  +  x^—x^  +  . 


l+x 
hence,  substituted  into  (24), 

log  {l+x)=  f{l-x+x^-x^  +  .  .  .)dx 

=  i  dx  -  i  xdx  +  j  xMx  -  j  x^dx  +  .  .  . 

C"      X^      J^ 

hence,  if  x  is  very  small,  —  is  negligible,   and,  therefore,  all 
terms  beyond  the  first  are  negligible,  thus, 
log  (1  +x)  =.r; 

while,  if  the  second  term  is  still  appreciable  in  value,  the  more 
complete,  but  still  fairly  simple  expression  can  be  used, 

log(l+j-)=x-^  =  .r(^i-^j 


64  ENGINEERING  MATHEMATICS. 

If  instead  of  the  natural  logarithm,  as  used  above,  the 
decimal  logarithm  is  required,  the  following  relation  may  be 
applied : 

logio  a  =  logio£  loge  a  =  0.4343  logs  a, 

logio  a  is  expressed  by  logs  a,  and  thus  (19),  (20)  (21)  assume 
the  form, 

/      x^    x^    x'*  \ 

logio  (l+x)=0.4343(^x--+-— ^+.  .  .  j 

or,  approximately, 

logio(l+.r)=0.4343x; 
or,  more  accurately, 

logio  (l+.r)=0.4343.r(l-|) 

B.    DIFFERENTIAL   EQUATIONS. 

49.  The  representation  by  an  infinite  series  is  of  special 
value  in  those  cases,  in  which  no  finite  expression  of  the  func- 
tion is  known,  as  for  instance,  if  the  relation  between  x  and  y 
is  given  by  a  differential  equation. 

Differential  equations  are  solved  by  separating  the  variables, 
that  is,  bringing  the  terms  containing  the  one  variable,  y,  on 
one  side  of  the  equation,  the  terms  with  the  other  variable  x 
on  the  other  side  of  the  equation,  and  then  separately  integrat- 
ing both  sides  of  the  equation.  Very  rarely,  however,  is  it 
possible  to  separate  the  variables  in  this  manner,  and  where 
it  cannot  be  done,  usually  no  systematic  method  of  solving  the 
differential  equation  exists,  but  this  has  to  be  done  by  trying 
different  functions,  until  one  is  found  which  satisfies  the 
equation. 

In  electrical  engineering,  currents  and  voltages  are  dealt 
with  as  functions  of  time.  The  current  and  e.m.f.  giving  the 
power  lost  in  resistance  are  related  to  each  other  by  Ohm's 
law.  Current  also  produces  a  magnetic  field,  and  this  magnetic 
field  by  its  changes  generates  an  e.m.f. — the  e.m.f.  of  self- 
inductance.  In  this  case,  e.m.f.  is  related  to  the  change  of 
current;  that  is,  the  differential  coefficient  of  the  current,  and 
thus  also  to  the  differential  coefficient  of  e.m.f.,  since  the  e.m.f. 


POTENTIAL  SERIES  AND  EXPONENTIAL  FUNCTION.     65 

is  related  to  the  current  by  Ohm's  law.  In  a  condenser,  the 
current  and  therefore,  b}'  Ohm's  law,  the  e.m.f.,  depends  upon 
and  is  proportional  to  the  rate  of  change  of  the  e.m.f,  impressed 
upon  the  condenser;  that  is,  it  is  proportional  to  the  differential 
coefficient  of  e.m.f. 

Therefore,  in  circuits  having  resistance  and  inductance, 
or  resistance  and  capacity,  a  relation  exists  between  currents 
and  e.m.fs.,  and  their  differential  coefficients,  and  in  circuits 
having  resistance,  inductance  and  capacity,  a  double  relation 
of  this  kind  exists;  that  is,  a  relation  between  current  or  e.m.f. 
and  their  first  and  second  differential  coefficients. 

The  most  common  differential  equations  of  electrical  engineer- 
ing thus  are  the  relations  between  the  function  and  its  differential 
coefficient,  which  in  its  simplest  form  is, 

dy 

i-y-: (26) 

or 

dy 

I-"y (27) 

and  where  the  circuit  has  capacity  as  well  as  inductance,  the 
second  differential  coefficient  also  enters,  and  the  relation  in 
its  simplest  form  is, 

d-y 


or 


dii-y' (28) 

dhi 

d-'^y (23) 

and  the  most  general  form  of  this  most  common  differential 
equation  of  electrical  engineering  then  is, 

g  +  2c|  +  a,  +  6  =  0 (30) 

The  differential  equations  (26)  and  (27)  can  easily  be  inte- 
grated by  separating  the  variables,  but  not  so  with  equations 
(28),  (29)  and  (30);   the  latter  are  preferably  solved  by  trial. 

50.  The  general  method  of  solution  may  be  illustrated  with 
the  equation  (26) : 

t-y (2«) 


66  ENGINEERING  MATHEMATICS. 

To  determine  whether  this  equation  can  be  integrated  by  an 
infinite  series,  choose  such  an  infinite  series,  and  then,  by  sub- 
stituting it  into  equation  (26),  ascertain  whether  it  satisfies 
the  equation  (26);  that  is,  makes  the  left  side  equal  to  the  right 
side  for  every  value  of  .r. 

Let, 

?/  =  ao+aiX  +  a2x2  +  a3X^4-a4-r^+ (31) 

be  an  infinite  series,  of  which  the  coefficients  rto,  a\,  ao,  as.  .  . 
are  still  unknown,  and  by  substituting  (31)  into  the  differential 
equation  (26),  determine  whether  such  values  of  these  coefficients 
can  be  found,  which  niake  the  series  (31)  satisfy  the  equation  (26). 
Differentiating  (31)  gives, 

■£  =  ai+2a2X  +  3a3X^+4:a^x^  + (32) 

The  differential  equation  (26)  transposed  gives, 

Substituting  (31)  and  (32)  into  (33),  and  arranging  the  terms 
in  the  order  of  x,  gives, 

(ai  —  tto)  +  {2a2—ai)x  +  (3a3  —  a2)x^ 

+  (4a4 -03)^3  +  (5a5 - a4)^* +  ..  .=0.     .     (34) 

If  then  the  above  series  (31)  is  a  solution  of  the  diiTerential 
equation  (26),  the  expression  (34)  must  be  an  identity;  that  is, 
must  hold  for  every  value  of  x. 

If,  however,  it  holds  for  every  value  of  x,  it  does  so  also 
for  a;  =  0,  and  in  this  case,  all  the  terms  except  the  first  vanish, 
and  (34)  becomes, 

ai  — ao  =  0;     or,     ai=c!o (35) 

To  make  (31)  a  solution  of  the  differential  equation  {a\  —  ao) 
must  therefore  equal  0.  This  being  the  case,  the  term  (ai  —  ao) 
can  be  dropped  in  (34),  which  then  becomes, 

(2a2~ai)x  +  {'^a3—a2)x^  +  {4a4—a3)x^  +  {i'ma—a4}x^  +  -  ■  -=0; 

or, 

X{{2a2-ai)  +  {3a3-a2)x  +  {4a4-a3)x^  +  .  .  .}=0. 


POTENTIAL  SERIES  AND  EXPONENTIAL  FUNCTION.     67 


Since  this  equation  must  hold  for  every  value  of  x,  the  second 
factor  of  the  equation  must  be  zero,  since  the  first  factor,  x,  is 
not  necessarily  zero.     This  gives, 

(2a2-ai)  +  (3a3-a2)2:  +  (4a4-03)x2  +  .  .  .=0. 

As  this  equation  holds  for  every  value  of  x,  it  holds  also  for 
x  =  0.  In  this  case,  however,  all  terms  except  the  first  vanish, 
and, 

2a2-ai=0; (36) 

hence, 

Oi 


02=2, 


and  from  (35), 


ax 


2 


Continuing  the  same  reasoning, 

Sos  — rt2  =  0,     4a4  — 03=0,  etc. 

Therefore,  if  an  expression  of  successive  powers  of  x,  such  as 
(34),  is  an  identity,  that  is,  holds  for  every  value  of  x,  then  all 
the  coefficients  of  all  the  powers  of  x  must  separately  he  zero* 

Hence, 

tti  — ao=0;     or     ai  =  rto; 

(i\     ao 
2a2—ai=0;    or    ^2  =  — =-:t; 


Sas  —  a2  =  0 ;    or 


2      2 
as     3      |3, 

4o4-a3  =  0;     or    04  =  — =  rj; 


etc., 


etc. 


(37) 


*  The  reader  must  realize  the  difference  between  an  expression  (34),  as 
equation  in  x,  and  as  substitution  product  of  a  function;  that  is,  as  an 
identity. 

Regardless  of  the  values  of  the  coefficients,  an  expression  (34)  as  equation 
gives  a  number  of  separate  values  of  x,  the  roots  of  the  equation,  which 
make  the  left  side  of  (34)  equal  zero,  that  is,  solve  the  equation.  If,  however, 
Vie  infinite  series  (31)  is  a  solution  of  the  differential  equation  (26),  then 
the  expression  (34),  which  is  the  result  of  substituting  (31)  into  (26),  must 
be  correct  not  only  for  a  limited  number  of  values  of  x,  which  are  the  roots 
of  the  equation,  but  for  all  values  of  x,  that  is,  no  matter  what  value  is 
chosen  for  x,  the  left  side  of  (34)  must  always  give  the  same  result,  0,  that 
is,  it  must  not  be  changed  by  a  change  of  x,  or  in  other  words,  it  must  not 
contain  x,  hence  all  the  coefficients  of  the  powers  of  x  must  be  zero. 


68 


ENGINEERING  MATHEMATICS. 


Therefore,  if  the  coefficients  of  the  peries  (31)  are  chosen 
by  equation  (37),  this  series  satisfies  the  differential  equation 
(26);   that  is, 

r         x^  x^  x^       1 

y  =  ao    l+a:+^+j3+p-  +  .  .  .|.     .     .     .     (38) 

is  the  solution  of  the  differential  equation, 

dy 

51.  In   the   same   manner,    the   differential   equation    (27), 


dz 


(39) 


is  solved  by  an  infinite  series, 

z  =  aQ+aix  +  a2X^+a3X^  +  .  . .,    ....     (40) 

and  the  coefficients  of  this  series  determined  by  substituting 
(40)  into  (39),  in  the  same  manner  as  done  above.     This  gives, 

(ai  —  aao)  +  {2a2  —  aai)x  +  (Sas  —  aa2)x'^ 

+  i4a4-aa3)x^  +  .  ..=0,  .     (41) 

and,  as  this  equation  must  be  an  identity,  all  its  coefficients 
must  be  zero;  that  is, 

ai  — aao  =  0;     or    ai  =  aao; 

a         c? 
2a2— aai=0;     or     02  =  «i  77  =  ^0  it; 


a         cr 
3^3  —  aa-z  =  0 ;     or     as  =  ^2  o  =  oo  rr  ; 

o  >j 

a         a!^ 
4         |4- 


(42) 


etc.,  etc. 

and  the  solution  of  differential  equation  (39)  is, 

[,  a2r2       ^3^.3       fjAj^A  > 

2  =  ao|l+aa:+— +-p-+-p-  +  ...|.      .     .     (43) 

52.  These  solutions,  (38)  and  (43),  of  the  difTerential  equa- 
tions (26)  and  (39),  are  not  single  solutions,  but  each  contains 
an  infinite  number  of  solutions,  as  it  contains  an  arbitrary 


POTENTIAL  SERIES  AND  EXPONENTIAL  FUNCTION.    69 

constant  ao',    that  is,  a  constant  whicli  may  have  any  desired 
numerical  vahie. 

This  can  easily  be  seen,  since,  if  2  is  a  solution  of  the  dif- 
ferential equation, 

dz 

then,  any  multiple,  or  fraction  of  z,  hz,  also  is  a  solution  of  the 
differential  equation; 

d{hz) 

since  the  h  cancels. 

Such  a  constant,  ao,  which  is  not  determined  by  the  coeffi- 
cients of  the  mathematical  problem,  but  is  left  arbitrary,  and 
requires  for  its  determinations  some  further  condition  in 
addition  to  the  differential  equation,  is  called  an  integration 
constant.  It  usually  is  determined  by  some  additional  require- 
ments of  the  'physical  problem,  which  the  differential  equation 
represents;  that  is,  by  a  so-called  terminal  condition,  as,  for 
instance,  by  having  the  value  of  y  given  for  some  particular 
value  of  ;r,  usually  for  x  =  0,  or  x=  oc. 

The  differential  equation, 

dx.=y' (^^) 

thus,  is  solved  by  the  function, 

y  =  aoyo, (45) 

where, 

'T"2        1*3       •T»4 

ya  =  l+x+^+^+'r^  +  ...,     ....     (46) 

and  the  difTerential  equation, 

dz 

Tx-"'' (*^^ 

is  solved  by  the  function, 

z  =  a(iZo, (48) 

where, 

a^x^     aH^    a'^x*  ,._. 

0o  =  l+ax  +  -2-+-j3-+-np  + -     -     (49) 


70  ENGINEERING  MATHEMATICS. 

yo  and  Zq  thus  are  the  simplest  forms  of  the  solutions  y  and  z 
of  the  differential  equations  (26)  and  (39). 

53-  It  is  interesting  now  to  determine  the  value  of  ?/".  To 
raise  the  infinite  series  (46),  which  represents  2/o>  to  the  nth 
power,  would  obviously  be  a  very  complicated  operation. 

However, 

Ay''  ,  dy  ^^, 

Ay 
and  since  from  (44)  w~^^' ^^^^ 

by  substituting  (51)  into  (50), 

^="»";    (52) 

hence,  the  same  equation  as  (47),  but  with  ?/"  instead  of  z. 
Hence,  if  y  is  the  solution  of  the  differential  equation, 

dy 

then  z^y^  is  the  solution  of  the  differential  equation   (52), 

dz 

-T-^nz. 

dz 

However,  the  solution  of  this  differential  equation  from  (47), 
(48),  and  (49),  is 

z  =  aoZ()', 


that  is,  if 


then, 


2o  =  l+nj+-2--t-j3— +. 


/v»2        'v»3 

2/o  =  l+^+2'+i3 +• 


Tn?x^    n^x^ 


2o  =  !/o"  =  l+^^+-^y-+-T3-  +  - •  •  ;   •    .     .     (53) 

therefore  the  series  y  is  raised  to  the  nth  power  by  multiply- 
ing the  variable  x  by  n. 


POTENTIAL  SERIES  AND  EXPONENTIAL  FUNCTION.     71 

Substituting  now  in  equation  (53)  for  n  the  value  —  gives 

1  111 

t/o*  =  l+l+2+n-  +  |j  +  --.  ;      ....     (54) 

that  is,  a    constant    numerical  value.      This  numerical  value 
equals  2.7182818.  . .,  and  is  usually  represented  by  the  symbol  e. 
Therefore, 

hence, 

/j*2         /y^         -^4 

^/^=cx  =  i+a;+-+p-+ij  +  ..., (55) 

and 

Jl'^X'^      Iv^jfi       Tl^X^ 

2o  =  ^o"  =  (s'^)"=s"''  =  l+^-^+-i7-+-Tr +"Tr  +  -  •  •  -     (^^) 

therefore,  the  infinite  series,  which  integrates  above  differential 
equation,  is  an  exponential  function  with  the  base 

£  =  2.7182818 (57) 

The  solution  of  the  differential  equation, 

t-y' (^) 

thus  is, 

t/  =  ao^^ (5^) 

and  the  solution  of  the  differential  equation, 

!-«!'. («") 


IS 


2/  =  ao£"^ (61) 


where  Qq  is  an  integration  constant. 

The  exponential  function  thus  is  one  of  the  most  common 
functions  met  in  electrical  engineering  problems. 

The  above  described  method  of  solving  a  problem,  by  assum- 
ing a  solution  in  a  form  containing  a  number  of  unknown 
coefficients,  Gq,  ai,  02  . . .,  substituting  the  solution  in  the  problem 
and  thereby  determining  the  coefficients,  is  called  the  method 
of  indeterminate  coefficients.     It  is  one  of  the  most  convenient 


72  ENGINEERING  MATHEMATICS. 

aiul    most    frequently    used    methods    of     solving   engineering 
problems. 

EXAMPLE   1. 

54.  In  a  4-pole  500-Yolt  50-kw.  direct-current  shunt  motor, 
the  resistance  of  the  field  circuit,  inclusive  of  field  rheostat,  is 
250  ohms.  Each  field  pole  contains  4000  turns,  and  produces 
at  500  volts  impressed  upon  the  field  circuit,  8  megalines  of 
magnetic  flux  per  pole. 

What  is  the  equation  of  the  field  current,  and  how  much 
time  after  closing  the  field  switch  is  required  for  the  field  cur- 
rent to  reach  90  per  cent  of  its  final  value? 

Let  r  be  the  resistance  of  the  field  circuit,  L  the  inductance 
of  the  field  circuit,  and  i  the  field  current,  then  the  voltage 
consumed  in  resistance  is, 

Cj.  =  ri. 

In  general,  in  an  electric  circuit,  the  current  produces  a 
magnetic  field;  that  is,  lines  of  magnetic  flux  surrounding  the 
conductor  of  the  current;  or,  it  is  usually  expressed,  interlinked 
with  the  current.  This  magnetic  field  changes  with  a  change  of 
the  current,  and  usually  is  proportional  thereto.  A  change 
of  the  magnetic  field  surrounding  a  conductor,  however,  gen- 
erates an  e.m.f,  in  the  conductor,  and  this  e.m.f.  is  proportional 
to  the  rate  of  change  of  the  magnetic  field;  hence,  is  pro- 
portional   to    the    rate    of    change    of    the    current,    or    to 

di 

■—,  with  a  proportionality  factor  L,  which  is  called  the  induct- 
ance of  the  circuit.     This  counter-generated  e.m.f.  is  in  oppo- 

di 
sition  to  the  current,   —L-r,,  and  thus  coasumes  an  e.m.f., 

di 
+  L-7:,  which  is  called  the  e.m.f.  consumed  by  self-inductance, 
dt' 

or  inductance  e.m.f. 

Therefore,  by  the  inductance,  L,  of  the  field  circuit,  a  voltage 

is  consumed  which  is  proportional  to  the  rate  of  change  of  the 

field  current,  thus, 

-  di 


POTENTIAL  SERIES  AND  EXPONENTIAL  FUNCTION.     73 

Since  the  supply  voltage,  and  thus  the  total  voltage  consumed 
in  the  field  circuit,  is  e  =  500  volts, 

.     ^  di 
e  =  ri+Lj^; (62) 

or,  rearranged, 

di     e—ri 
JC~L~' 

Substituting  herein, 

M==e— n; (63) 

hence, 

du  di 

dt^~^It' 
gives, 

du         r 

Tr~v (64) 

This  is  the  same  differential  equation  as  (39),  with  a=— y, 

Li 

and  therefore  is  integrated  by  the  function, 

u  =  aos.    I-    ; 
therefore,  resubstituting  from  (63), 

and 

1  = t   "- (65) 

r      r  ^ 

This  solution  (65),  still  contains  the  unknown  quantity  ao; 
or,  the  integration  constant,  and  this  is  determined  by  know- 
ing the  current  i  for  some  particular  value  of  the  time  /. 

Before  closing  the  field  switch  and  thereby  impressing  the 
voltage  on  the  field,  the  field  current  obviously  is  zero.  In  the 
moment  of  closing  the  field  switch,  the  current  thus  is  still 
zero;    that  is, 

(•  =  0  for  /  =  0.  .     (66; 


74  ENGINEERING   MATHEMATICS. 

Substituting  these  values  in  (65)  gives, 

0  = -;    or    ao  =  +e, 

r     r 

hence, 

i=:^(i-r^')  ......  (67) 

is  the  final  solution  of  the  differential  equation  (62);  that  is, 
it  is  the  value  of  the  field  current,  i,  as  function  of  the  time,  t, 
after  closing  the  field  switch. 

After  infinite  time,  t  =  QC,  the  current  i  assumes  the  final 
value  io,  which  is  given  by  substituting  ^  =  oo  into  equation 
(67),  thus, 

e     500     ^  ._-,. 

io  =  -  =  -:5-r7:  =  2  amperes;       ....     (o8) 

hence,  by  substituting  (68)  into  (67),  this  equation  can  also  be 
written, 

=  2(l-rr'), (69) 

where  ao  =  2  is  the  final  value  assumed  by  the  field  cui-rent. 
The  time  ti,  after  which  the  field  current  i  has  reached  90 
per  cent  of  its  final  value  ig,  is  given  by  substituting  i  =  0.9iQ 
into  (69),  thus, 

o.9?-o=Vo(i-r^"), 

and 

ri'"=o.i. 

Taking  the  logarithm  of  both  sides, 

r 
-j;^!  log  c-=-l; 

and 

«i=-r^ (70) 

rlog  £ 


POTENTIAL  SERIES  AND  EXPONENTIAL  FUNCTION.    75 

55.  The  inductance  L  is  calculated  from  the  data  given 
in  the  problem.  Inductance  is  measured  by  the  number  of 
interlinkages  of  the  electric  circuit,  with  the  magnetic  flux 
produced  by  one  absolute  unit  of  current  in  the  circuit;  that 
is,  it  equals  the  product  of  magnetic  flux  and  number  of  turns 
divided  by  the  absolute  current. 

A  current  of  ^=2  amperes  represents  0.2  absolute  units, 
since  the  absolute  unit  of  current  is  10  amperes.  The  number 
of  field  turns  per  pole  is  4000;  hence,  the  total  number  of  turns 
n  =  4x4000  =  16,000.  The  magnetic  flux  at  full  excitation, 
or  tQ=0.2  absolute  units  of  current,  is  given  as  <?=8xl0^  lines 
of  magnetic  force.     The  inductance  of  the  field  thus  is: 

I^^^^^Q^^.f^^OWxlO^  absolute  units=6W, 

Iq  0.2 

the  practical  unit  of  inductance,  or  the  henry  {h)  being  10^ 
absolute  units. 

Substituting  L  =  640  r  =  250  and  e  =  500,  into  equation  (67) 
and  (70)  gives 

i  =  2(l-£-0-39'), 

and 

Therefore  it  takes  about  6  sec.  before  the  motor  field  has 
reached  90  per  cent  of  its  final  value. 

The  reader  is  advised  to  calculate  and  plot  the  numerical 
values  of  ^  from  equation  (71),  for 
t  =  0,  0.1,  0.2,  0.4,  0.6,  0.8,  1.0,  1.5,  2.0,  3,  4,  5,  6,  8,  10  sec. 

This  calculation  is  best  made  in  the  form  of  a  table,  thus; 


and, 

hence, 

and. 


£""-^^'  =  A'-0.39nog£, 
log£    =0.4343; 
O.SQflog,    =0.1694<; 


76 


ENGINEERING  MA  THEM  A  TICS. 


The  values  of  £-0  39<  pg^,^  ^i^^  ^^^  taken  directly  from  the 
tables  of  the  exponential  function,  at  the  end  of  the  book. 


t 

0.1694< 

-0.1694/ 

.-0.39< 

j_ --0.39< 

t  = 

2(1- £--39') 

=^  A'  -  0. 1 594/ 

0.0 
0.1 
0.2 
0.4 
0.6 
0.8 
etc. 

0 
0.0170 
0.0339 
0 . 0678 
O.IOIG 
0.1355 

0 
0.9830-1 
0.9661-1 
0.9322-1 
0.8984-1 
0.8645-1 

1 

0 .  962 
0 .  925 
0.855 
0.791 
0.732 

0 
0.03S 
0.075 
0.145 
0.209 
0.268 

0 

0.076 
0.150 
0.290 
0.418 
0.536 

EXAMPLE    2. 

56.  A  condenser  of  20  mf.  capacity,  is  charged  to  a  potential 
of  Cq  =  10,000  volts,  and  then  discharges  through  a  resistance 
of  2  megohms.     What  is  the  equation  of  the  discharge  current, 

and  after  how  long  a  time  has 
the  voltage  at  the  condenser 
dropped  to  0.1  its  initial  value? 
A  condenser  acts  as  a  reser- 
voir of  electric  energy,  similar 
to  a  tank  as  water  reservoir. 
If  in  a  water  tank.  Fig.  27,  A 
is  the  sectional  area  of  the  tank, 
e,  the  height  of  water,  or  water 
pressure,  and  water  flows  out 
of  the  tank,  then  the  height  e 
decreases  by  the  flow  of  water; 
that  is  the  tank  empties,  and 
the  current  of  water,  i,  is  proportional  to  the  change  of  the 

de 
water  level  or  height  of  water,  — ,  and   to  the  area  A  of  the 

tank;  that  is,  it  is. 


Fig.  27.     Water  Reservoir. 


2  = 


A 


df 


(72) 


The  minus  sign  stands  on  the  right-hand  side,  as  for  positive 
i;  that  is,  out-flow,  the  height  of  the  water  decreases;  that  is, 
de  is  negative. 


POTENTIAL  SERIES  AND  EXPONENTIAL  FUNCTION.     77 

In  an  electric  reservoir,  the  electric  pressure  or  voltage  e 
corresponds  to  the  water  pressure  or  height  of  the  water,  and 
to  the  storage  capacity  or  sectional  area  A  of  the  water  tank 
corresponds  the  electric  storage  capacity  of  the  condenser, 
called  capacity  C.  The  current  or  flow  out  of  an  electric 
condenser,  thus  is, 

--4^ (^3> 

The  capacity  of  condenser  is, 

C  =  20  mf  =  20  X 10-  «  farads. 
The  resistance  of  the  discharge  path  is, 
r  =  2Xl06  ohms; 
hence,  the  current  taken  by  the  resistance,  r,  is 

.     e 


r 


and  thus 


-^  dt     r' 
and 

dt~     Cr^- 
Therefore,  from  (GO)  (Gl), 


and  for  t  =  0,  e  =  6'o  =  10,000  ''olts;  hence 

10,000  =  ao, 


and 

0.1  of  the  initial  value: 
Is  reached  at: 


e  =  e^,e    Cr 


=  10,000c-" "25e  ^oj^g.  ^     (74) 

e  =  0.1eo, 

Cr 
ti=-, =  92  sec.    ......     (75) 


78  ENGINEERING  MATHEMATICS. 

The  reader  is  advised  to  calculate  and  plot  the  numerical 
values  of  e,  from  equation  (74),  for 

^  =  0;  2;  4;  6;  8;  10;  15;  20;  30;  40;  GO;  80;  100;  150;  200  sec. 

57-  Wherever  in  an  electric  circuit,  in  addition  to  resistance, 
inductance  and  capacity  both  occur,  the  relations  between 
currents  and  voltages  lead  to  an  equation  containing  the  second 
diiferential  coefficient,  as  discussed  above. 

The  simplest  form  of  such  equation  is: 

d^^^'y (^^) 

To  integrate  this  by  the  method  of  indeterminate  coefficients, 
we  assume  as  solution  of  the  equation  (76)  the  infinite  series, 

y  =  aQ+a\X-{-a2X^ -\-a-}X^  ■^-a^x'^-^ (77) 

in  which  the  coefficients  ao,  a\,  ao,  «3,  04.  .  .  are  indeterminate. 
Differentiating  (77)  twice,  gives 

--|  =  2a2-i-2x3a3X  +  3x4a4x2+4X5a5>r3  +  .  .  .  ,  .  (78) 

and  substituting  (77)  and  (78)  into  (76)  gives  the  identity, 

2a2+2x3a3X+3x4a4X-'+4x5a5x3  +  .  .  . 

=a(ao+aiX+a2X^+a3X^  +  .  .  .); 

or,  arranged  in  order  of  x, 

(2a2  — aao)  +J'(2x3a3  — aai)  +x2(3x4a4— aa2) 

+  x3(4x5a5-aa3)+.  • -  =  0 (79) 

Since  this  equation  (79)  is  an  identity,  the  coefficients  of 
all  powers  of  x  must  individually  equal  zero.  This  gives  for 
the  determination  of  these  hitherto  indeterminate  coefficients 
the  equations, 

2a2  —  aao  =  0 ; 
2X3a3-aai=0; 
3x4a4  — aa2  =  0; 
4  X  5a5  —  aaz  =  0,  etc. 


POTENTIAL  SERIES  AND  EXPONENTIAL  FUNCTION.     79 
Therefore 

ci2  =  -:j-',  a3== 


li' 


aa2      OoCt^  aa.-j      flifl^ 

«4  =  :7T77  =  -nr-;  «5  =  43<5==l5"' 


3X4 

|4 

aa^ 

ttga^ 

5X6 

'  l« 

aae 

aott^ 

7X8       |8   '  "     8X9       |9 

etc.,  etc. 

Substituting  these  values  in  (77), 

In  this  case,  two  coefficients  ao  and  ai  thus  remain  inde- 
terminate, as  was  to  be  expected,  as  a  differential  equation 
of  second  order  must  have  two  integration  constants  in  its 
most  general  form  of  solution. 

Substituting  into  this  equation, 

62  =  a; 
that  is, 

b  =  \/a,       (81) 

ih-'^'y'-  ■ («2) 


and 


62^2        64^.4        ^6j~6 


^       b^x^     h^x^     67,r7 

6x+-^  +  ^^+-T^  +  ...    .     (83) 


80  EXGIXEERIXG   MATHEMATICS. 

In  this  case,  instead  of  the  integration  constants  ao  and  ai, 
the  two  new  integration  constants  A  and  B  can  be  introduced 
by  the  equations 

ao  =  A+B    and     -^  =  A-B; 

0 

hence,  ^^  ^i 

A  =  — ^ —    and     B  =  ^^ — , 
and,  substituting  these  into  equation  (83),  gives, 

f  ^2^.2       53j;3       &4^4  1 

,7   r  ^'^"  ^^-^^  ?>^-^^      1 

+  5|l-6x+-T^ [3" +l4"~ +•••[•     •     ^^^^ 

The  first    series,  how^ever,  from  (56),  for  /i  =  6  is  e'^^',  and 
the  second  series  from  (56),  for  n=  —6  is  e"^*. 
Therefore,  the  infinite  series  (83)  is, 

ij  =  Ae^^''+Be-^^; (85) 

that  is,  it  is  the  sum  of  two  exponential  functions,  the  one  with 
a  positive,  the  other  with  a  negative  exponent. 
Hence,  the  differential  equation, 

d7^  =  «^' (^^^ 

is  integrated  by  the  function, 

y-^Ae-^^'+Bs-f'^, f86) 

where, 

h  =  \'a. (87) 

However,  if  a  is  a  negative  quantity,  b=\\i  is  imaginary, 
and  can  be  represented  by 

b  =  jc, (88) 

where 

c^=-a (89) 

In  this  case,  equation  (86)  assumes  the  form, 


POTENTIAL  SERIES  AND  EXPONENTIAL  FUNCTION.     81 

that  is,  if  in  the  differential  equation  (76)  a  is  a  positive  quantity, 
=  +b^,  this  differential  equation  is  integrated  by  the  sum  of 
the  two  exponential  functions  (86) ;  if,  however,  a  is  a  negative 
quantity,  =  —c^,  the  solution  (86)  appears  in  the  form  of  exponen- 
tial functions  with  imaginary  exponents  (90). 

58.  In  the  latter  case,  a  form  of  the  solution  of  differential 
equation  (76)  can  be  derived  which  does  not  contain  the 
imaginary  appearance,  by  turning  back  to  equation  (80),  and 
substituting  therein  a==  —c'^,  which  gives. 


(91) 


c^x^ 


4x4 


y  =  ao\  1 — 9-  + 


IT 


6  r6 


C'r 


+  . 


c^r 


ai        _ 
c  i  ''•'       13 


i5  r5 


+  ■ 


C^X 


■-+.  . 


or,  writing  A  =  ao  and  5  =  — 


ai 


y=^A\  1 — :r-+- 


,6,6 


C"X 


+ 


5r5 


„  I  Cx-"       C^X 

+  B\  c-i---|3-+-|^ 


+  , 


(92) 


The  solution  then  is  given  by  the  sum  of  two  infinite  series, 
thus, 

/■6,6 


and 


as 


,      ,       ,       C2x2        C^X* 

u{cx)=i — 2~^ir 


6"-l' 


+ 


y\0  y*0 


5r5 


C^X 


v{cx)=CX--r:^+-r^ 


y  =  Au(cx)  +Bv{cx). 


+  . 


(03) 


(94) 


In  the  w-series,  a  change  of  the  sign  of  x  does  not  change 
the  value  of  u, 

u{  —  cx)=u{+cx) (95) 


Such  a  function  is  called  an  even  function. 


82  ENGINEERING  MATHEMATICS. 

In  the  t'-scries,  a  change  of  the  sign  of  x  reverses  the  sign 
of  V,  as  seen  from  (93): 

v{-cx)  =  -v{  +  ci) (96) 

Such  a  function  is  called  an  odd  function. 
It  can  be  shown  that 

w(cx)  =  coscx    and     r (ex)  =  sin  car;    .     .     .     (97) 

hence, 

i/  =  x4  cos  cx+5  sin  ex, (98) 

where  A  and  B  are  the  integration  constants,  which  are  to  be 
determined  by  the  terminal  conditions  of  the  physical  problem. 
Therefcre,  the  solution  of  the  differential  equation 

^  =  «^>        (99) 

has  two  diiTerent  forms,   an  exponential  and  a  trigonometric. 
If  a  is  positive 

d^2-+b% (100) 

it  is: 

y  =  A£  +  ^  +  Be-^^,        (101) 

If  a  is  negative, 

^2  =  -c'y, (102) 

it  is: 

7/  =  ^  cos  ex +  5  sin  ex (103) 

In  the  latter  case,  the  solution  (101)  would  appear  as   ex- 
ponential function  with  imaginary  exponents; 

^  =  ^£  +  K^+5£-''^*  (104) 

As  (104)  obviously  must  be  the  same  function  as  (103),  it 
follows  that  ex])oncntial  functions  with  imaginary  exponents 
must  be  expressible  by  trigonometric  functions. 


POTENTIAL  SERIES  AND  EXPONENTIAL  FUNCTION.    83 


59.  The  exponential  functions  and  the  trigonometric  func- 
tions, according  to  the  preceding  discussion,  are  expressed  by 
the  infinite  series, 


.r^     J'*     x*     x^ 

£-  =  l+Xf-+p-+ij+-+, 
Z        o        4        0 

^•2       •v-4       v-6 

COSX=l  —  '—+'r-r  —  'TX  +  —.  .   . 
2         4         O 


rrO  'yO  yi 

sin  X  =  X  —  T77+nr  —  ^  + 
1 


(105) 


Therefore,  substituting  /w  for  x, 


£'"  =  1+;m- 


w^      .  w^     ?r 


.  M^       U^         "7 


-m+Mr+J^-|7T-JTT+- 


"^      -'13    '  14    '  MS      16      ^  u 


ii-"     ir    n^ 


-['-l+\i-m  + 


u-^     u^     u' 

o       0       7 


However,  the  first  part  of  this  series  is  cos  u,  the   latter  part 
sin  u,  by  (105);    that  is, 


£^"  =  cos  M+/sin  u. 


Substituting  —u  for  +u  gives, 

£-?"  =  eos  u—j  sin  u. 
Combining  (106)  and  (107)  gives, 

£  +  }■"    -(_     £-JU 


(106) 


(107) 


cos  u  = 


2 


and 


ff+j"—  f-j" 


sin  u  = 


-^J 


(108) 


Substituting  in  (106)  to  (108),  jv  for  u,  gives, 
e  ~  "  =  cos  /v  4-  y  sin  jv,  ] 
£"^''  =  cos   jv  —  j  sin  p  j 


and, 


(109) 


84  ENGINEERING  MATHEMATICS. 


Adding  and  subtracting  gives  respectively, 

....     (110) 


cosjv  =  - — y^ — , 


and 


sin  ]V  = :z-. — 

By  these  equations,  (106)  to  (110),  exponential  functions 
with  imaginary  exponents  can  be  transformed  into  trigono- 
metric functions  with  real  angles,  and  exponential  functions 
with  real  exponents  into  trignometric  functions  with  imaginary 
angles,  and  inversely. 

Mathematically,  the  trigonometric  functions  thus  do  not 
constitute  a  separate  class  of  functions,  but  may  be  considered 
as  exponential  functions  with  imaginary  angles,  and  it  can  be 
said  broadly  that  the  solution  of  the  above  differential  equa- 
tions is  given  by  the  exponential  function,  but  that  in  this 
function  the  exponent  may  be  real,  or  may  be  imaginary,  and 
in  the  latter  case,  the  expression  is  put  into  real  form  by  intro- 
ducing the  trigonometric  functions. 

EXAMPLE    1. 

6o.  A  condenser  (as  an  underground  high-potential  cable) 
of  20  mf.  capacity,  and  of  a  voltage  of  eo  =  10,000,  discharges 
through  an  inductance  of  50  mh.  and  of  negligible  resistance* 
What  is  the  equation  of  the  discharge  current? 

The  current  consumed    by  a  condenser  of  capacity  C  and 

potential  difference  e  is    proportional  to  the  rate  of  change 

of  the  potential  difference,  and  to  the  capacity;    hence,  it  is 

de 
C  — ,  and  the  current   from  the   condenser;    or    its  discharge 
dt 

current,  is 

--4: ("" 

The  voltage  consumed  by  an  inductance  L  is  proportional 
to  the  rate  of  change  of  the  current  in  the  inductance,  and  to  the 
inductance ;   hence, 

-4 ("2) 


POTENTIAL  SERIES  AND  EXPONENTIAL  FUNCTION.     85 

Differentiating  (112)  gives, 

de       (Pi 

and  substituting  this  into  (111)  gives, 

t=-CL-,:    or,    ^=-^^,      .     •     .     (113) 
as  the  differential  equation  of  the  problem. 

This  equation  (113)  is  the  same  as  (102),  for  c^  =  j7j,  thus 
is  solved  by  the  expression, 

{  =  i4.cos      '. —  +Bsin — =-,        .     .     .     (114) 
y/LC  VLC 

and  the  potential  difference  at  the  condenser  or  at  the  inductance 
is,  by  substituting  (114)  into  (112), 

^\Bcoi>—L=-Asm-i=\.      .     (115) 

These  equations  (114)  and  (115)  still  contain  two  unknown 
constants,  A  and  B,  which  have  to  be  determined  by  the  terminal 
conditions,  that  is,  by  the  known  conditions  of  current  and 
voltage  at  some  particular  time. 

At  the  moment  of  starting  the  discharge;  or,  at  the  time 
^  =  0,  the  current  is  zero,  and  the  voltage  is  that  to  which  the 
condenser  is  charged,  that  is,  i  =  0,  an.l  e  =  eo. 

Substituting  these  values  in  equations  (114)  and  (115) 
gives, 


?o=^^5; 


0  =  A     and     e„ 
hence 

'r 

and,  substituting  for  A  and  B  the  values  in  (114)  and   (115), 

gives 

Ic   .       t 

o\L        vCL' 

and 

t 
e  =  en  cos  — ;^==^ 
""        VCL 


^  =  ^o\lT' 


(116) 


86  ENGINEERING  MATHEMATICS. 

Substituting  the   numerical  values,  Cq  =  10,000  volts,  C= 20 
mf.  =  20X10- 6  farads,  L  =  50  mh.  =0.05h.  gives, 


4 


y=0.02     and     vC'L  =  10-3; 


hence, 

1  =  200  sin  1000  t    and     e  =  10,000  cos  1000  t. 

6i.  The  discharge  thus  is  alternating.  In  reality,  due  to 
the  unavoidable  resistance  in  the  discharge  path,  the  alterna- 
tions gradually  die  out,  that  is,  the  discharge  is  oscillating. 

The  time  of  one  complete  period  is  given  by, 

1000/0=2;:;     or,     to^^^- 
Hence  the  frenquency, 

,     1      1000     ,,^       , 

f=-r  =  '^ —  =  159  cvcles  per  second. 

■^      to       27:  "  ^ 

As  the  circuit  in  addition  to  the  inductance  necessarily 
contains  resistance  r,  besides  the  voltage  consumed  by  the 
inductance  by  equation  (112),  voltage  is  consumed  by  the 
resistance,  thus 

er  =  ri, (117) 

and  the  total  voltage  consumed  by  resistance  r  and  inductance 
L,  thus  is 

di 


e^ri  +  L- (118) 


Differentiating  (118)  gives, 

de       di     -.  dH  .^^^. 

jr'-jt^^dP '"") 

and,  substituting  this  into  equation  (111),  gives. 

^+"^^^5+^^^^" (12"' 

as  the  differential  equation  of  the  problem. 

This  differential  equation  is  of  the  more  general  form,  (30), 
62.  The  more  general  differential  equation  (30). 

g  +  2^  +  a,  +  6  =  0, (121) 


POTENTIAL  SERIES  AND  EXPONENTIAL  FUNCTION.     87 
can,  by  substituting, 


which  gives 


2/+-  =  ^, (122) 


dy    dz 
dx    dx 


be  transformed  into  the  somewhat  simpler  form, 

d^z     ^  dz            ^  /-,«o\ 

-+2cJ^^a^  =  0 (123) 

It  may  also  be  solved  by  the  method  of  indeterminate 
coefficients,  by  substituting  for  z  an  infinite  series  of  powers  of 
X,  and  determining  thereby  the  coefficients  of  the  series. 

As,  however,  the  simpler  forms  of  this  equation  were  solved 
by  exponential  functions,  the  applicability  of  the  exponential 
functions  to  this  equation  (123)  may  be  directly  tried,  by  the 
method  of  indeterminate  coefficients.  That  is,  assume  as  solu- 
tion an  exponential  function, 

z  =  Ae^', (124) 

where  A  and   h  are  unknown  constants.     Substituting   (124) 
into  (123),  if  such  values  of  A  and  b  can  be  found,  which  make 
the  substitution  product  an  identity,    (124)   is  a  solution  of 
the  differential  equation  (123). 
From  (124)  it  follows  that, 

^  =  bAe^-;    and     ^=62^£6x    _         _     q25^ 
dx  dx^ 

and  substituting  (124)  and  (125)  into  (123),  gives, 

^£fcxj^2+2c5+a!=0 (12G) 

As  seen,  this  equation  is  satisfied  for  every  value  of  x,  that 
is,  it  is  an  identity,  if  the  parenthesis  is  zero,  thus, 

62+2c&+a  =  0, (127) 

and  the  value  of  6,  calculated  by  the  quadratic  equation  (127), 
thus  makes  (124)  a  solution  of  (123),  and  leaves  A  still  undeter- 
mined, as  integration  constant. 


88  ENGINEERING  MATHEMATICS. 

From  (127), 

h=  —c±  \''c'^  —  a] 
or,  substituting, 

Vc'-a  =  p, (128) 

into  ('128),  the  equation  becomes, 

b=-c±p (129) 

Hence,  two  values  of  b  exist, 

hi=—c  +  p      and      62=— c— p, 
and,  therefore,  the  differential  equation, 

is  solved  by  .4£'"^;   or,   by   As'"^,  or,   by  any  combination  of 
these  two  solutions.     The  most  general  solution  is, 

that  is, 

^  [  .      .     .     (131) 

=  £-^^lAi£+P^  +  yl2^"^'^f--. 

a 

As  roots  of  a  quadratic  equation,  61  and  62  may  both  be 
real  quantities,  or  may  be  complex  imaginary,  and  in  the 
latter  case,  the  solution  (131)  appears  in  imaginary  form,  and 
has  to  be  reduced  or  modified  for  use,  so  as  to  eliminate  the 
imaginary  appearance,  by  the  relations  (106)  and  (107). 

EXAMPLE  2. 

63.  Assume,  in  the  example  in  paragraph  60,  the  discharge 
circuit  of  the  condenser  of  C  =  20  mf.  capacity,  to  contain, 
besides  the  inductance,  L  =  0.05  h,  the  resistance,  r=125  ohms. 

The  general  equation  of  the  problem,  (120),  dividing  by 
C  L,  becomes, 

(Pi     r  di      i      ^  ,,_^. 


POTENTIAL  SERIES  AND  EXPONENTIAL  FUNCTION.     89 
This  is  the  equation  (123),  for: 

F=2500: 

(133) 


x  =  t,     2c  =  7=2500 

z  =  i,      a  =  -jTj~  =  10® 

If 

p==Vc^  —  a,     then 

//  r  y     1 

~2W        \" 

and,  writing 

p=- 


2L' 


and  since 


^  =  10     and     ^  =  2500, 

.'?  =  75     and      p  =  750.    J 
The  equation  of  the  current  from  (131)  then  it 


r"^^'  +  .4..£~2Z'  ^ 


J  •  J 


(134) 


(135) 


(136) 


(137) 


Tliis  equation  still  contains  two  unknown  quantities,  the  inte- 
gration constants  A^  and  A2,  which  are  determined  by  the 
terminal  condition:  The  values  of  current  and  of  voltage  at  the 
beginning  of  the  discharge,  or  ^  =  0. 

This  requires  the  determination  of  the  equation  of  the 
voltage  at  the  condenser  terminals.  This  obviously  is  the  voltage 
consumed  by  resistance  and  inductance,  and  is  expressed  by 
equation  (118), 

di 


e  =  ri~\-L 


at' 


(118) 


90  ENGINEERING  MATHEMATICS. 

hence,  substituting  herein  the  vahic  of  i  and  -r.,  from  equation 
(137),  gives 


=  r     2L  . 


i^^A.^-^^i'+V^^^-"^'! (138) 


and,  substituting  the  numerical  values   (133)  and   (136)  into 
equations   (137)  and  (138),  gives 


(139) 


and, 

e=100Ai£-5oot^25A2£-2ooo' 

At  the   moment  of  the  beginning  of  the  discharge,   ^  =  0, 
the  current  is  zero  and  the  voltage  is  10,000;  that  is, 

i  =  0;  1  =  0;  e  =  10,000 (140) 

Substituting  (140)  into  (139)  gives, 

0  =  .4i+A2,     10,000  =  100.4 1+ 25.42 ; 
hence, 

.42= -Ar.     .4 1  =  133.3;     .42= -133.3.    .     .     (141) 

Therefore,  the  current  and  voltage  are, 

i  =  133.3ic-5oo'-£-2ooo«j^       j 


e  =  13,333c--5oo'-3333£--ooo'  J 


(142) 


The  reader  is  advised  to  calculate  and  plot  the  numerical 
values  of  i  and  e,  and  of  their  two  components,  for, 

f  =  0,  0.2,  0.4,  0.6,  1,  1.2,  1.5,  2,  2.5,  3,  4,  5,  OxlO-^  sec. 


POTENTIAL  SERIES  AND  EXPONENTIAL  FUNCTION.     91 

64.  Assuming,  however,  that  the  resistance  of  the  discharge 
circuit  is  only  r  =  80  ohms  (instead  of  125  ohms,  as  assumed 
above) : 

4L 

r^— —  m  equation  (134)  then  becomes  —3600,  and  there- 
fore: 

s  =  \/-3600  =  60  V^  =  60y, 
and 

The  equation  of  the  current  (137)  thus  appears  in  imaginary 
form, 

{  =  £-800/j^^j  +  600j<_^^2£-600jO  _       _       _       (^43) 

The  same  is  also  true  of  the  equation  of  voltage. 

As  it  is  obvious,  however,  physically,  that  a  real  current 
must  be  coexistent  with  a  real  e.m.f.,  it  follows  that  this 
imaginary  form  of  the  expression  of  current  and  voltage  is  only 
apparent,  and  that  in  reality,  by  substituting  for  the  exponential 
functions  with  imaginary  exponents  their  trigononetric  expres- 
sions, the  imaginary  terms  must  eliminate,  and  the  equation 
(143)  appear  in  real  form. 

According  to  equations  (106)  and  (107), 

£+6ooj7_eos  600f+ysin  600^;  1 

£-  6ooj«  =  cos  600t  -  j  sin  600^  |  •   •     •     • 

Substituting  (144)  into  (143)  gives, 

t  =  £-8oo<j 5^  cos  600^+^2  sin  600^1,      .     .     (145) 

where  5i  and  B2  are  combinations  of  the  previous  integration 
constants  Ai  and  A  2  thus, 

B,  =  A,+A2,    and    52  =  y(^i-^2).   •     .      (146) 

By  substituting  the  numerical  values,  the  condenser  e.m.f., 
given  by  equation  (138),  then  becomes, 

e  =  £-800'!  (40  +30j)^i(cos  600^  +/  sin  6000 

+  (40-30jM2(cos  600«-y  sin  6000 1 
=  £~8oo«j  (405i  +3052)cos  600i  +  (4052-3050  sin  600t\.     (147) 


(144) 


92  ENGINEERING  MATHEMATICS. 

Since  for  t=0,  i  =  0  and  e  =  10,000  volts  (140),  substituting 
into  (145)  and  (147), 

0  =  5i  and  10,000  =  40  5i+30  B2. 
Therefore,  5i=0  and  52  =  333  and,  by  (145)  and  (147), 

i  =  333£-8oo'sin  000^;  1 

.   .     (148) 
e=  10,000s -«oo'  (cos  600  <  +  1.33  sin  6000- J 

As  seen,  in  this  case  the  current  i  is  larger,  and  current 
and  e.m.f.  are  the  product  of  an  exponential  term  (gradually 
decreasing  value)  and  a  trigonometric  term  (alternating  value); 
that  is,  they  consist  of  successive  alternations  of  gradually 
decreasing  amplitude.  Such  functions  are  called  oscillating 
functions.  Practically  all  disturbances  in  electric  circuits 
consist  of  such  oscillating  currents  and  voltages. 


600/  =  2r  gives,  as  the  time  of  one  complete  period, 
and  the  frequency  is 


T  =  -^  =  0.0105sec.; 
bOO 


f=7n=  95.3  cycles  per  sec. 

In  this  particular  case,  as  the  resistance  is  relatively  high, 
the  oscillations  die  out  rather  rapidly. 

The  reader  is  advised  to  calculate  and  plot  the  numerical 
values  of  i  and  e,  and  of  their  exponential  terms,  for  every  30 

T       T       T 

degrees,  that  is,  for  /  =  0,  jx,  2-^^,  3  y;^,  etc.,  for  the  first  two 

periods,  and  also  to  derive  the  equations,  and  calculate  and  plot 
the  numerical  values,  for  the  same  capacity,  C  =  20  mf.,  and 
same  inductance,  L  =  0.05/i,  but  for  the  much  lower  resistance, 
r  =  20  ohms. 

65.  Tables  of  s'^^  and  £~*,  for  5  decimals,  and  tables  of 
log  £■•"*  and  log  s"^,  for  6  decimals,  are  given  at  the  end  of 
the  book,  and  also  a  table  of  e"^  for  3  decimals.  For  most 
engineering  purposes  the  latter  is  sufficient;  where  a  higher 
accuracy  is  required,  the  5  decimal  table  may  be  used,  and  for 


POTENTIAL  SERIES  AND  EXPONENTIAL  FUNCTION.     93 

highest  accuracy  interpolation  by  the  logarithmic  table  may  be 
employed.     For  instance, 

£-13.6847^? 

From  the  logarithmic  table, 

log  £-10  =5.657055, 
log  £-3  =8.697117, 
log  £-0-6  =9.739423, 
log  £-0.08  =9.905256, 
log  £-0  004'' =  9.997959, 

f  interpolated, 

1  between  log  £-000^=9.998263, 
added  i  and  log  £-ooo5         =  9.997829), 

log  £- 13-68^^  =  4.056810  =0.056810  -  6. 
From  common  logarithmic  tables, 

^-13.0847^;^   1397QXlQ-6_ 

Note.  In  mathematics,  for  the  base  of  the  natural  loga- 
rithms, 2.718282  .  .  . ,  is  usually  chosen  the  symbol  e.  Since, 
however,  in  engineering  the  symbol  e  is  universally  used  to 
rejjresent  voltage,  for  the  base  of  natural  logarithms  has  been 
chosen  the  symbol  £,  as  the  Greek  letter  corresponding  to  e, 
and  £  is  generally  used  in  electrical  engineering  calculations  in 
this  meaning. 


CHAPTER  III. 
TRIGONOMETRIC  SERIES. 

A.   TRIGONOMETRIC   FUNCTIONS. 

66.  For  the  engineer,  and  especially  the  electrical  engineer, 
a  perfect  familiarity  with  the  trigonometric  functions  and 
trigonometric  formulas  is  almost  as  essential  as  familiarity  with 
the  multiplication  table.  To  use  trigonometric  methods 
efficiently,  it  is  not  sufficient  to  understand  trigonometric 
formulas  enough  to  be  able  to  look  them  up  when  required, 
but  they  must  be  learned  by  heart,  and  in  both  directions;  that 
is,  an  expression  similar  to  the  left  side  of  a  trigonometric  for- 
mula must  immediately  recall  the  right  side,  and  an  expression 
similar  to  the  right  side  must  immediately  recall  the  left  side 
of  the  formula. 

Trigonometric  functions  are  defined  on  the  circle,  and  on 
the  right  triangle. 

Let  in  the  circle.  Fig.  28,  the  direction  to  the  right  and 
upward  be  considered  as  positive,  to  the  left  and  downward  as 
negative,  and  the  angle  a  be  counted  from  the  positive  hori- 
zontal OA,  counterclockwise  as  positive,  clockwise  as  negative. 

The  'projector  s  of  the  angle  a,  divided  by  the  radius,  is 
called  sin  a;  the  projection  c  of  the  angle  a,  divided  by  the 
radius,  is  called  cos  a. 

The  intercept  t  on  the  vertical  tangent  at  the  origin  A, 
divided  by  the  radius,  is  called  tan  a;  the  intercept  ct  on  the 
horizontal  tangent  at  B,  or  90  deg.,  behmd  A,  divided  by  the 
radius,  is  called  cot  a. 


(1) 


94 


Thus,  in  Fig.  28, 

s 
r' 

c 
cos  a  =  -', 
r 

tan  a:  =  -; 

ct 
cot  a  =  — . 
r 

TRIGONOMETRIC  SERIES. 


95 


In  the  right  triangle,  Fig.  29,  with  the  angles  a  and  /?, 
opposite  respectively  to  the  cathetes  a  and  b,  and  with  the 
hypotenuse  c,  the  trigonometric  functions  are: 


sina  =  cos /9  =  -;     cosa:=sin^  =  - 


tan  a  =  cot /?  =  r;    cot  a=tan /3  =  — . 
^     0  a 


-.      .     .     (2) 


By  the  right  triangle,  only  functions  of  angles  up  to  90  deg., 

or    -,  can  be  defined,  while  by  the  circle  the  trigonometric 

functions  of  any  angle  are  given.     Both  representations  thus 
must  be  so  familiar  to  the  engineer  that  he  can  see  the  trigo- 


FiG.  28.     Circular  Trigonometric 
Functions. 


Fig.  29.     Triangular  Trigono- 
metric Functions. 


nometric  functions  and  their  variations  with  a  change  of  the 
angle,  and  in  most  cases  their  numerical  values,  from  the 
mental  picture  of  the  diagram. 

67.  Signs  of  Functions.  In  the  first  quadrant,  Fig.  28,  all 
trigonometric  functions  are  positive. 

In  the  second  quadrant.  Fig.  30,  the  sin  a  is  still  positive, 
as  s  is  in  the  upward  direction,  but  cos  a  is  negative,  since  c 
is  toward  the  left,  and  tan  a  and  cot  a  also  are  negative,  as  / 
is  downward,  and  d  toward  the  left. 

In  the  third  quadrant,  Fig.  31,  sin  a  and  cos  a  are  both 


96 


ENGINEERING  MA  THEM  A  TICS. 


negative:   s  being  downward,  c  toward  the  left;   but  tan  a  and 
fot  a  are  again  positive,  as  seen  from  t  and  ct  in  Fig.  31. 


Fig.  30.     Second  Quadrant. 


Fig.  31.    Third  Quadrant. 


In  the  fourth  quadrant,  Fig.  32,  sin  a  is  negative,  as  s  is 
downward,  but  cos  a  is  again  positive,  as  c  is  toward  the  right; 

tan  a  and  cot  a  are  both 
negative,  as  seen  from  t  and 
ct  in  Fig.  32. 

In  the  fifth  quadrant  all 
the  trigonometric  functions 
again  have  the  same  values 
as  in  the  first  quadrant,  Fig. 
28,  that  is,  360  deg.,  or  27r, 
or  a  multiple  thereof,  can  be 
added  to,  or  subtracted  from 
the  angle  a,  without  changing 
the  trigonometric  functions, 
but  these  functions  repeat 
after  every  360  deg.,  or  2;r; 


\              ct 

- 

B 

1  c           1 

A  1 

\  ° 

t 

Fig.  32.     Fourth  Quadrant. 


that  is,  have  2;r  or  360  deg.  as  their  period. 


SIGNS   OF  FUNCTIONS 


Function. 

Positive. 

Negative. 

sin  a 
cos  a 
tan  a 
cot  a 

1st  and  2d 
1st  and  4th 
1st  and  3d 
1st  and  3d 

3d    and  4th  quadrant 
2d    and  3d 
2d    and  4th          " 
2d    and  4th 

(3) 


TRIGONOMETRIC  SERIES.  97 

68.  Relations  between  sin  a  and  cos  a.     Between  sin  «  and 
cos  a  the  relation, 

sin^  a -j-cos^  a  =  l, (4) 

exists;  hence, 


sin  a  =  Vl  —  cos^  a ;  ,.  . 
(4a) 

cos  «  =  v^l  — sin^' <v.   J 

Equation  (4)  is  one  of  those  which  is  frequently  used  in 
both  directions.  For  instance,  1  may  be  substituted  for  the 
sum  of  the  squares  of  sin  a  and  cos  a,  while  in  other  cases 
sin^  a  +cos^  a  may  be  substituted  for  1.     For  instance, 

1         sin^a  +  cos^a      /sin«\- 

— ^= 5 =    +I=tan2a  +  1. 

cos'^  a  cos-  a  \  cos  a/ 

Relations  between  Sines  and  Tangents. 

sin  a     "I 


tan  a  =  ■ 

cos  a 

cos  a 

cot  n=-. 

sin  a 

hence 

1 

cot  a  = 


(5) 


tan  a  = 


tan  <t ' 

1 

cot  a' 


(5a) 


As  tan  a  and  cot  a  are  far  less  convenient  for  trigonometric 
calculations  than  sin  a  and  cos  a,  and  therefore  are  less  fre- 
quently applied  in  trigonometric  calculations,  it  is  not  neces- 
sary to  memorize  the  trigonometric  formulas  pertaining  to 
tan  a  and  cot  a,  but  where  these  functions  occur,  sin  a  and 
and  cos  a  are  substituted  for  them  by  equations  (5),  and  the 
calculations  carried  out  with  the  latter  functions,  and  tan  a 
or  cot  a  resubstituted  in  the  final  result,  if  the  latter  contains 

sin  «         .  . 

,  or  its  reciprocal. 

cos  a 

In  electrical  engineering  tan  a  or  cot  a  frequently  appears 

as  the  starting-point  of  calculation  of   the  phase  of  alternating 

currents.     For  instance,  if  «  is  the  phase    angle  of  a  vector 


98  ENGINEERING  MATHEMATICS. 

quantity,  tan  a  is  given  as  the  ratio  of  the  vertical  component 
over  the  horizontal  component,  or  of  the  reactive    component 
over  the  power  component. 
In  this  case,  if 

tan  «  =  -J- , 

0 

sin  g  =    ,  ,      and      cos  a=    ,  :      .     (56) 

or,  if 

cot  a  =  -T , 
a 

sin  q:=  '  ,      and      cos  a=    ,  .     .     (5c) 

The  secant  functions,  and  versed  sine  functions  are  so 
little  used  m  engineering,  that  they  are  of  interest  only  as 
curiosities.     They  are  defined  by  the  following  equations: 

1 


sec  a 


cosec  a 


cos  a 
1 


sin  a 

sin  vers  «  =  1  —  sin  a, 

cos  vers  a  =  1  —  cos  a. 

69.  Negative  Angles.  From  the  circle  diagram  of  the 
trigonometric  functions  follows,  as  shown  in  Fig.  33,  that  when 
changing  from  a  positive  angle,  that  is,  counterclockwise 
rotation,  to  a  negative  angle,  that  is,  clockwise  rotation,  s,  t, 
and  ct  reverse  their  direction,  but  c  remains  the  same;  that  is. 


sin  (— a)  =  —sin  a,   ■ 
cos  ,(— a)  =  +C0S  a, 
tan  {—a)  =  —tan  a, 
cot  (  —  a)  =  —cot  a, 


(6) 


COS  a  thus  is  an  "  even  function,"  while  the  three  others  are 
"  odd  functions." 


TRIGONOMETRIC  SERIES. 


99 


Supplementary  Angles.  From  the  circle  diagram  of  the 
trigonometric  functions  follows,  as  shown  in  Fig.  34,  that  by 
changing  from  an  angle  to  its  supplementary  angle,  s  remains 
in  the  same  direction,  but  c,  t,  and  ct  reverse  their  direction, 
and  all  four  quantities  retain  the  same  numerical  values,  thus, 


sin  (n—a)  =  +sin  a, 
cos  (tt— a)  = —cos  a, 
tan  {7t—a)  =  —tan  «, 
cot  {7t—a)  =  —  cot  a. 


(7) 


Fig.  33.  Functions  of  Negative  Fig.  34.  Functions  of  Supplementary 

Angles.  Angles. 

Complementary   Angles.     Changing  from  an  angle  a  to  its 

complementary  angle  90° —  «,  or  .y  — «,  as  seen  from  Fig.  35, 

the  signs  remain  the  same,  but  s  and  c,  and  also  t  and  d  exchange 
their  numerical  values,  thus, 

1 


sin^|-aj 


cos  a. 


cosi  y  —  «■ )  =sm  a, 


tan(  y  —  a  I  =cot  «, 


cot 


s-« 


--tan  a. 


(S) 


100 


ENGINEERING  MA  THEM  A  TICS. 


70.  Angle  (a±-).     Adding,  or  subtracting  -  to  an  angle  a, 
gives  the  same  numerical  values  of  the  trigonometric  functions 

1/ 


Fig.  35.     Functions  of  Complemen-     Fig.  36.     Functions  of  Angles  Plus 
tary  Angles.  or  Minus  -. 

as  a,  as  seen  in  Fig.  36,  but  the  (Urection  of  s  and  c  is  reversed, 
while  t  and  ct  remain  in  the  same  direction,  thus, 
sin  (a:±-)  =  —sin  «,  ] 

cos  («  ±-)  =  -cos  a,  I 

!•  ■  •     (9) 

tan  (a-  ±;r)  =  +tan  a, 

cot  (a  ±7:)=  +cot  a. 


\       ct 

B        ct 

y 

ff\ 

\ 

t 

I            c      0 

\       ^ 

J 

\ 

A      •" 
t 

\        ct 

h 

B         ct 

y 

A 

y 

\ 

t 

1              0 

\  1 

1 

\ 

A    ' 
t 

Fig.  37.  Functions  of  Angles+  -^.       Fig.  38.  Functions  of  Angles  Minus  ^, 

Angle(a±^  j.     Adding  ^,  or  90  dog.  lo  an  angle  a,  inter- 
changes the  functions,  s  and  c,  and  i  and  d,  and  also  reverses 


TRIGONOMETRIC  SERIES. 


101 


the  direction  of  the  cosine,  tangent,  and  cotangent,  but  leaves 
the  sine  in  the  same  direction,  since  the  sine  is  positive  in  the 
second  quadrant,  as  seen  in  Fig.  37. 

Subtracting  7^,  or  90  deg.  from  angle  a,  interchanges  the 

functions,  s  and  c,  and  t  and  ct,  and  also  reverses  the  direction, 
except  that  of  the  cosine,  which  remains  in  the  same  direction; 
that  is,  of  the  same  sign,  as  the  cosine  is  positive  in  the  first 
and  fourth  quadrant,  as  seen  in  Fig.  38.      Therefore, 


+  COS  (V, 


sin 

(       ^\ 

COS 

tan 

(«4) 

cot 

sin 

COS 

tan 

r"2J 

cot 

(-0 

(10) 


=  —  sm  a, 

=  —cot  a, 

--^  —tan  a, 
=  —cos  a, 

=  +sin  a', 

^  —cot  tt', 

=  —tan  a. 


Numerical   Values.     From  the  circle  diagram,  Fig.  28,  etc., 
follows  the  numerical  values : 


.     .     (11) 


sill 

0°  =  0 

sin 

30°  =  i 

sin 

45°  =  ^^   2 

sin 

60°  =  i  .    3 

sin 

90°  =  1 

sin 

120°  =  J^\    3 

etc. 

cos      0°=1 
cos    30°  =  i\ '3 
cos    45°  =  -t\  '2 
cos    60°  =  i 
cos    90°  =  0 
cos  120°=-^ 
etc. 


tan      0°  =  0 
tun    45°=  1 
tan    90°=  CO 
tan  135°= -1 
etc. 


cot      0°  =  oo 
cot    45°  =  1 
cot    90°  =  0 
cot  135°= -1 
etc. 


(12) 


102 


ENGINEERING  MATHEMATICS. 


). 


(13) 


71.  Relations  between  Two  Angles.     The  following  relations 
are  developed  in  text-books  of  trigonometry : 

sin  {a  +^)  =sin  a  cos  /3  +  cos  a  sin  ^, 

sin  (a  — ^5)  =sin  a  cos  ,5  — cos  a  sin  [i, 

cos  {a  +/?)  =cos  a  cos  ^9  — sin  a  sin  /9, 

cos  (a  — ^)=cosa  cos  /5+sin  a  sin  /?, 

Herefrom    follows,  by  combining    these    equations    (13)    in 
pairs : 

cos  a  cos  /5  =  i|cos  (a  +^)  +cos  (a  —  /?) }, 

sin  a  sin  /3  =  ^{cos  (a  — ^5)  — cos  («+/?}), 

sin  a  cos/?  =  i!sin  (a:+/'?)+sin  («  —  /?)!, 

cos  a  sin|9  =  ijsin  («+/?)  — sin  (a  — ,5)|. 

By  substituting  ai  for  («+/?),  and  /?i  for  (a  — /3)  in  these 
equations  (14),  gives  the  equations, 

sm  «!  +sm  ,81  =     2  sm  — :z cos 


(14) 


sin  ai  — sin  ,/?i 


2  sm  — :t —  cos 


cos  ai+cos /3i=     2  cos 


2 


cos 


COS  ai  — COS  /?i 


2  sm — ::: —  sm 


2 
2 
2 

«!  — ^1. 


(15) 


These  three  sets  of  equations  are  the  most  important  trigo- 
nometric formulas.  Their  memorizing  can  be  facilitated  by 
noting  that  cosine  functions  lead  to  products  of  equal  func- 
tions, sine  functions  to  products  of  unequal  functions,  and 
inversely,  products  of  equal  functions  resolve  into  cosine, 
products  of  unequal  functions  into  sine  functions.  Also  cosine 
functions  show  a  reversal  of  the  sign,  thus:  the  cosine  of  a 
sum  is  given  by  a  difference  of  products,  the  cosine  of  a  differ- 
ence by  a  sum,  for  the  reason  that  with  increasing  angle 
the  cosine  function  decreases,  and  the  cosine  of  a  sum  of  angles 
thus  would  be  less  than  the  cosine  of  the  single  angle. 


TRIGONOMETRIC  SERIES.  103 

Double  Angles.    From  (13)  follows,  by  substituting  a  for  /?: 

sin  2a  =  2  sin  a  cos  a, 
cos  2a  =  cos2  a  — sin^  a, 
=  2  cos^  oc  —  1, 
=  1  —  2  sin^  a. 
Herefrom  follow 


(16) 


1  — cos  2a  ,  „         1+ COS  2a 

sin^  a  = and      cos^  «  = 


2 


(16a) 


72.  Differentiation. 


d   ,.      . 

-J-  (sm  a)=  +COS  a, 
da 


-5-  (  cos  a)=  —sin  a. 
da 


(17) 


The  sign  of  the  latter  differential  is  negative,  as  with  an 
increase  of  angle  a,  the  cos  a  decreases. 


Integration. 


I  sin  ada=  —cos a, 

I  cos  ada=  +sina. 
Herefrom  follow  the  definite  integrals: 

sin  {a+a)da  =  0;  ! 

[.      .     .     . 

I        cos{a  +  a)da  =  0; 
Jc  i 

I        sin  (a +  a)da;  =  2  cos  (c  + a);      ] 

Xc  +  n  I 

COS  (a+a)da:= -2sin  (c+a);  1 


(18) 


(I80) 


(186) 


104 


ENGINEERING  MATHEMATICS. 


C 

2 

f 


sin  ada  =  0 ; 


_n.l 


COS  ada  =  0 ; 


(18c) 


sin  ada  =  + 


J 


cos  ada  =  +1. 


(18d) 


73.  Binomial.  One  of  the  most  frequent  trigonometric 
operations  in  electrical  engineering  is  the  transformation  of  the 
binomial,  a  cos  a +6  sin  a,  into  a  single  trigonometric  function, 
by  the  substitution,  a  =  c  cos  p  and  6  =  c  sin  p;  hence, 


where 


a  cas  a-Vh  sin  «  =  c  cos  (a  —  /)), 


c  =  \- a-  +  ^-'     antl     tan  /)  =  — ;    . 


or,  by  the  transformation,  a  =  c  sin  g  and  h  =  c  cos  q, 

a  cos  a +  &  sin  a  =  c  sin  («+</),    .     . 
where 

r CL 

c  =  \^d^-\-h'^     and     tan^  =  r-.    •     • 
74.  Polyphase  Relations. 


(19) 

(20) 

(21) 
(22) 


S 


I  cos    a -t-a±- 


n 
2mi- 


=  0, 


^J  sin  (  a  +  a  ±  ^^^- —  I  =  0, 


(23) 


where  m  and  n  are  integer  numbers. 

Proof.     The  points  on  the  circle  which   defines   the   trigo- 

/             2mi-\ 
nometric  function,   by  Fig.   28,   of  the  angles  {a-\-a± I, 


TRIGONOMETRIC  SERIES. 


105 


are  corners  of  a  regular  polygon,  inscribed  in  the  circle  and 
therefore  having  the  center  of  the  circle  as  center  of  gravity. 
For  instance,  for  n  =  7,  m  =  2,  they  are  shown  as  Pi,  Po,  .  ■  .  P7, 
in  Fig.  39.  The  cosines  of  these  angles  are  the  projections  on 
the  vertical,  the  sines,  the  projections  on  the  horizontal  diameter, 
and  as  the  sum  of  the  projections  of  the  corners  of  any  polygon, 


R. 

\ 

B 

P5 

^  «> 

\ 

I 

V. 

\ 

P 

S3    / 

3 

'■\ 

/ 

V 

Fig.  39.     Polyphase  Relations. 


F)G.  40.     Triangle. 


on  any  line  going  through  its  center  of  gravity,  is  zero,  both 
sums  of  equation  (23)  are  zero. 

■^         /  2mi7z\         I       ,      2miTi\     n 

2_i  cos  [a+a± I  cos  (  a:  +  0 ± I  =  :j  cos  (a -- 6), 


2mi7:\     n         ,       ,. 


(24) 


>^     .     /               2mi7:\    .     / 
y  i  sm  (  a  ^-a± I  sin  I 

^    .     /              2mi7z\         I       ^      2mir\     n    .      .       ,^ 
>j  sin  I  a  +a± )  cos  I  «  +o± )  =:t  sin  {a—n). 


These  equations  are  proven  by  substituting  for  the  products 
the  single  functions  by  equations  (14),  and  substituting  them 
in  equations  (23). 

75.  Triangle.  If  in  a  triangle  a,  .5,  and  j-  are  the  angles, 
opposite  respectively  to  the  sides  a,  b,  c,  Fig.  40,  then, 


sin  a  -^ sin  /3 -f- sin  y  =  a-i-h^c, 


.     (25) 


106  ENGINEERING  MATHEMATICS. 

a2  +  62_c2 


cos  X 


2ab       ' 
or 

(^  =  a~  +  b"  —  2ab  cos  y. 

.  ab  sin  r 

Area  =  ^ — ^ — - 

c^  sin  a  sin  13 
2  sin  7- 


.     (26) 


(27) 


B.   TRIGONOMETRIC   SERIES. 

76.  Engineering  phenomena  usually  are  either  constant, 
transient,  or  periodic.  Constant,  for  instance,  is  the  terminal 
voltage  of  a  storage-battery  and  the  current  taken  from  it 
through  a  constant  resistance.  Transient  phenomena  occur 
during  a  change  in  the  condition  of  an  electric  circuit,  as  a 
change  of  load;  or,  disturbances  entering  the  circuit  from  the 
outside  or  originating  in  it,  etc.  Periodic  phenomena  are  the 
alternating  currents  and  voltages,  pulsating  currents  as  those 
produced  by  rectifiers,  the  distribution  of  the  magnetic  flux 
in  the  air-gap  of  a  machine,  or  the  distribution  of  voltage 
around  the  commutator  of  the  direct-current  machine,  the 
motion  of  the  piston  in  the  steam-engine  cylinder,  the  variation 
of  the  mean  daily  temperature  with  the  seasons  of  the  year,  etc. 

The  characteristic  of  a  periodic  function,  y=f(x),  is,  that 
at  constant  intervals  of  the  independent  variable  x,  called 
cycles  or  periods,  the  same  values  of  the  dependent  variable  y 
occur. 

Most  periodic  functions  of  engineering  are  functions  of  time 
or  of  space,  and  as  such  have  the  characteristic  of  univalence; 
that  is,  to  any  value  of  the  independent  variable  x  can  corre- 
spond only  one  value  of  the  dependent  variable  y.  In  other 
words,  at  any  given  time  and  given  point  of  space,  any  physical 
phenomenon  can  have  one  numerical  value  only,  and  therefore 
must  be  represented  by  a  univalent  function  of  time  and  space. 

Any  univalent  periodic  function, 

y=M,       (1) 


TRIGONOMETRIC  SERIES.  107 

can  be  expressed  by  an  infinite  trigonometric  series,  or  Fourier 
series,  of  the  form, 

y  =  ao+ai  cos  cx+a2  cos  2cx  +  a3  cos  Scx  +  .  .  .  . 

+  61  sin  cx  +  bosin  2CX  +  63  sin  3c.r  +  .  .  .    :       ....     (2) 

or,  substituting  for  convenience,  cx-=0,  this  gives 

y  =  ao+ai  cos  6+a2  cos  26  +  a3  cos  3^  +  .  .  . 

+61  sin  <9 +62  sin  2^+63  sin  3^  +  .  ..  ; (3) 

or,  combining  the  sine  and  cosine  functions  by  the  binomial 

(par.  73), 

y  =  ao+ci  cos  (d-j^i)  +C2COS  {26-^2)  +C3 cos(3^-/33)  +.  .  .  1 
=  ao+cisin(^  +  n)+C2sin(2^  +  r2)+C3sin  (3^  +  r3) +•  •  .  T  ^  ^ 

where 


tan/9„  =  -^; 

On 

or  tan  r,i=7— • 

On 


(5) 


The  proof  hereof  is  given  by  showing  that  the  coefficients 
a„  and  6  „  of  the  series  (3)  can  be  determined  from  the  numencal 
values  of  the  periodic  function  (1),  thus, 

y=m=W) (6) 

Since,  however,  the  trigonometric  function,  and  therefore 
also  the  series  of  trigonometric  functions  (3)  is  univalent,  it 
follows  that  the  periodic  function  (6),  y=fo{d),  must  be  uni- 
valent, to  be  represented  by  a  trigonometric  series. 

77.  The  most  important  periodic  functions  in  electrical 
engineering  are  the  alternating  currents  and  e.m.fs.  Usually 
they  are,  in  first  approximation,  represented  by  a  single  trigo- 
nometric function,  as : 

i  =  7o  cos  {d—cu); 
or, 

e  =  eo  sin  {d—d); 

that  is,  they  are  assumed  as  sine  waves. 


108  ENGINEERING  MATHEMATICS. 

Theoretically,  obviously  this  condition  can  never  be  perfectly 
attainetl,  and  frequently  the  deviation  from  sine  shape  is  suffi- 
cient to  require  practical  consideration,  especially  in  those  cases, 
where  the  electric  circuit  contains  electrostatic  capacity,  as  is 
for  instance,  the  case  with  long-distance  transmission  lines, 
underground  cable  systems,  high  potential  transformers,  etc. 

However,  no  matter  how  much  the  alternating  or  other 
periodic  wave  differs  from  simple  sine  shape — that  is,  however 
much  the  wave  is  "  distorted,"  it  can  always  be  represented 
by  the  trigonometric  series  (3). 

As  illustration  the  following  apj^lications  of  the  trigo- 
nometric series  to  engineering  problems  may  be  considered: 

{A)  The  determination  of  the  equation  of  the  periodic 
function;  that  is,  the  evolution  of  the  constants  a„  and  h^^  of 
the  trigonometric  series,  if  the  numerical  values  of  the  periodic 
function  are  given.  Thus,  for  instance,  the  wave  of  an 
alternator  may  be  taken  by  oscillograph  or  wave-meter,  and 
by  measuring  from  the  oscillograph,  the  numerical  values  of 
the  periodic  function  are  derived  for  every  10  degrees,  or  every 
5  degrees,  or  every  degree,  depending  on  the  accuracy  required. 
The  problem  then  is,  from  the  numerical  values  of  the  wave, 
to  determine  its  equation.  While  the  oscillograph  shows  the 
shape  of  the  wave,  it  obviously  is  not  possible  therefrom  to 
calculate  other  quantities,  as  from  the  voltage  the  current 
under  given  circuit  conditions,  if  the  wave  shape  is  not  first 
represented  by  a  mathematical  expression.  It  therefore  is  of 
importance  in  engineering  to  translate  the  picture  or  the  table 
of  numerical  values  of  a  periodic  function  into  a  mathematical 
expression  thereof. 

(B)  If  one  of  the  engineering  quantities,  as  the  e.m.f.  of 
an  alternator  or  the  magnetic  flux  in  the  air-gap  of  an  electric 
machine,  is  given  as  a  general  periodic  function  in  the  form 
of  a  trigonometric  series,  to  determine  therefrom  other  engineer- 
ing quantities,  as  the  current,  the  generated  e.m.f..  etc. 

A.  Evaluation  of  the  Constants  of  the  Trigonometric  Series  from 
the  Instantaneous  Values  of  the  Periodic  Function. 

78.  Assuming  that  the  numerical  values  of  a  univalent 
periodic  function  y=fo{G)  are  given;  that  is,  for  every  value 
of  6,  the  corresponding  value  of  y  is  known,  either  by  graphical 
representation,  Fig.  41;    or,  in  tabulated  form,  Table  I,  but 


TRIGONOMETRIC  SERIES. 


109 


the  equation  of  the  periodic  function  is  not  known.     It  can  be 
represented  in  the  form, 

y  =  aQ  +  ai  cos  ^+a2Cos  26  +  a3  cos  3^-"-.  .  .  +an  cosnd  +  .  .  . 

+  61  sin  ^+62  sin  2(9  +  63  sin  3^-;  .  .  .+6„sinn^-f .  .  .  ,    (7) 

and  the  problem  now  is,  to  determine    the  coefficients  ao,  ai, 
(1-2  •••  b\,  ho  ...  . 


Fig.  41.     Periodic  Functions. 
TABLE    I. 


e 

y 

e 
90 

+  50 

e 

y 

e 

y 

+  85 

0 

-60 

180 

+  122 

270 

10 

-49 

100 

+  61 

190 

+  124 

280 

+  65 

20 

-38 

110 

+  71 

200 

+  126 

290 

+  35 

30 

-26 

120 

+  81 

210 

+  125 

300 

+  17 

40 

-12 

130 

+  90 

220 

+  123 

310 

0 

50 

0 

140 

+  99 

230 

+  120 

320 

-13 

60 

+  11 

150 

+  107 

240 

+  116 

330 

-26 

70 

+  27 

160 

+  114 

250 

+  110 

340 

-38 

80 

+  39 

170 

+  \v.i^ 

260 

+  100 

350 

-49 

90 

+  50 

180 

+  122 

270 

+  85 

360 

1 

-60 

Integrate  the  equation  (7)  between  the  Umits  0  and  2r: 

J  I  ~\dd  =  a^)\''d0^ax  \  co^  Odd  +  ao  \  ~\os  2ddd  + .  .  . 
0  Jo  Jo  Jo 

a„  I      cos  n/9d^  +  .  .  .+61  j     ^\\\  ddO -\- 


+  ( 


+  62  I     sin  2^rf^  +  .  ..+?>„  I     ^mnddd^ 


■r 


■f'* 


/  0 

cos  nd  /2' 


/cos  2d  Z^"  _       _,      /cos  nd  I 
I       2/0        ■  "       V      n     /o 


+  , 


no  ENGINEERING  MATHEMATICS. 

All  the  integrals  containing  trigonometric  functions  vanish, 
as  the  trigonometric  function  has  the  same  value  at  the  upper 
limit  2-  as  at  the  lower  limit  0,  that  is, 

/cos  nd  J^"     I ,       ^  ^s     ^ 

/ /     =-(cos  2n7r— cos  0)=0; 

/sin  nd  /s-     1  •    an     a 

/ /     =-(sm  2?i7r  — sin  0)=0, 

/       n    /q        n 

and  the  result  is 

\'\jd0^aJo/'^=2r.aQ: 
Jo  /      /  0 

hence 

ao  =  J-r'V^^ (8) 

ydd  is  an  element  of  the  area  of  the  curve  y,    Fig.  41,  and 

ydd  thus    is  the  area  of   the  periodic    function  y,  for   one 

0  _ 

period;  that  is, 

aQ  =  ^A, (9) 

where  ^  =  area  of  the  periodic  function  y=^fo{0),  for  one  period; 

that  is,  from  ^  =  0to  6'  =  2r. 

A 
2r  is  the  horizontal  width  of  this  area  A,  and  -pr-  thus  is 

the  area  divided  b}'^  the  width  of  it;  that  is,  it  is  the  average 
height  of  the  area  A  of  the  periodic  function  y:  or,  in  other 
words,  it  is  the  average  value  of  y.     Therefore, 

ao  =  avg.  (y)o~^ (10) 

The  first  coefficient,  oo,  thus,  is  the  average  value  of  the 
instantaneous  values  of  the  periodic  function  y,  between  0  =  0 
and  0  =  2-. 

Therefore,  averaging  the  values  of  y  in  Table  I,  gives  the 
first  constant  qq. 

79.  To  determine  the  coefficient  a„,  multiply  equation  (7) 
by  cos  nO,  and  then  integrate  from  0  to  2;r,  for  the  purpose  of 
making  the  trigonometric  functions  vanish.     This  gives 


TRIGONOMETRIC  SERIES.  Ill 

p.  rir.  r2r. 

I     y  cos  nOdQ  =  ao  I     cos  nOdO  ^a\  I      cos  n./9  cos  ^(/^  + 
Jo  Jj  Jo 

r2r.  r2n 

+  a2  I     cosn^cos2^d^  +  .  .  .+a„  I     cos^  nddO  +  ... 
Jo  Jo 

+  61  I     cosnds\nOdO+b2\      QOiindiiin20dd  +  .  .  . 
Jo  Jo 

+  b„  I     cos  nd  sin  nOdO  +  .  .  . 
Jo 

Henco,   by  the   trigonometric   equations   of  the   preceding 
section : 
r2K  rin  rin 

I      ?/cosn^rf^=ao  I      cosn^rf^+ai)    ^[cos(n+l)^+cos(7i-l)^]d^ 
Jo  Jo  Jo 

i[cos  (n+2)^+cos  {n-2)Q'\dO ^ .  .  . 

X2k 
^{l+cos2n6)dd  +  .  .  . 

+  ?)i  j  "'i[sin  (n  +  l)^-sin  in-l)d]dd 

/'2rr 

+62  I     i[sin  (n  +  2)^-sin  {n-2)d]dd  +  .  .  . 
Jo 
r'2K 

+b„\     hsm2nddd  +  .  .  . 

Jo 

All  these  integrals  of  trigonometric  functions  give  trigo- 
nometric functions,  and  therefore  vanish  between  the  limits  0 
and  2;r,  and  there  only  remains  the  first  term  of  the  integral 
multiplied  with  a„,  which  does  not  contain  a  trigonometric 
function,  and  thus  remains  finite : 

an  I      7ydO  =  aJ^j    =ann, 

and  therefore, 

r2r. 

I     y  cos  nOdd^ttnT^', 

hence 

1   p'^ 
0'n  =  -  I     y  COS  nddd (11) 

^Jo 


112  ENGINEERING  MATHEMATICS. 

If  the  instantaneous  values  of  y  are  multiplied  with  cos  nd, 
and  the  product  ^„  =  2/cos  nO  plotted  as  a  curve,  y  cos  nddd  is 
an  element  of  the  area  of  this  curve,  shown  for  n  =  3  in  Fig.  42, 

and  thus    I     y  cos  nGdd  is  the  area  of  this  curve;  that  is, 
Jo 

an  =  -A„, (12) 


Fig.  42.     Curve  of  y  cos  3d. 

where  An  is  the  area  of  the  curve  y  cos  nd,  between  6  =  0  and 
6  =  27:. 

As  2r  is  the  width  of  this  area  A„,  —  is  the  average  height 

of  this  area;   that  is,  is  the  average  value  of  y  cos  nO,  and  -An 
thus  is  twice  the  average  value  of  y  cos  n6;  that  is, 

an  =  2  avg.  (?/ cos  71^)0"" (13) 


Fig.  43.     Curve  of  y  sin  3^. 

The  coefficient  a„  of  cos  n6  is  derived  by  multiplying  all 
the  instantaneous  values  of  y  by  cos  n6,  and  taking  twice  the 
average  of  the  instantaneous  values  of  this  product  y  cos  n6. 


+  a2  j     sin  n^  cos  2/9d^  +  .  .  .  4-a„  I      sm  nd  cos  nddd +  ..  . 
Jo  Jo 

sin  nd  sin  OdO  +  62  I     sin  nd  sin  2/9d^  + .  . 
+  6„  I     sin2?i/9d^  + 


TRIGONOMETRIC  SERIES.  113 

80.  6„  is  determined  in  the  analogous  manner  by  multiply- 
ing y  by  sin  nd  and  integrating  from  0  to  2;r;  by  the  area  of  the 
curve  y  sin  nd,  shown  in  Fig.  43,  for  n  =  3, 

r'27:  r2K  /^2r. 

I     y  sin  nddO  =  ao  (     sin  nOdQ  +  Oi  |      sin  nd  cos  ^d^ 
^0  Jo  Jo 

rir.  rzn 

>|      sm  n^  cos  2/9d^  +  .  .  .  4-a„  I      si 
Jo  Jo 

s'mnd  siaOdO  -\-b2  |     s 

=  ao  I     shi  nddO  +  ai  I     ^sin  {n  -:-l)^+sin  (n-l)d]dd 
Jo  Jo 

+  02  I  '"i[sin  (n  +  2)^  +  sin  (n-2)0]dd  +  .  .  . 

+a„  (     ^sin  2n^J/?  + .  .  . 
Jo 

+  61  j      i[cos  (??.-  l)^-cos  {ji  +  l)d]dd 

+62  I    "Mcos  (n- 2) (9 -cos  (n  +  2)^]d<9  +  .  .  . 

+  b„\'  h['^-cos2nO]dO  +  ... 
Jo 

—  h,A     IdO^hnTz; 
Jo 
hence, 

1  r-" 

6„  =  _        yi^mnOdO (14) 

^Jo 

=  i.1,/, (15) 

where  A,/  is  the  area  of  the  curve  ?/,/  =  ^  sin  nft.     Hence, 

6„  =  2  avg.  (|/sinn^)o"'',       (16) 


114 


ENGINEERING  MA  THEM  A  TICS. 


and  the  coefficient  of  sin  nO  thus  is  derived  by  multiplying  the 
instantaneous  values  of  y  with  sin  nd,  and  then  averaging,  as 
twice  the  average  oi  ysm  nO. 

8i.  Any  univalent  periodic  function,  of  which  the  numerical 
values  y  are  known,  can  thus  be  expressed  numerically  by  the 
equation, 
y  =  aQ  +  ai  cos  d  +  a2  cos  26  +  .  .  .  +  a„  cos  nd  + .  .  . 

+  hism  0+h2sm20  +  .  .  .+hns'mnO  +  .  .  .  ,     .     (17) 

where  the  coefficients  Qq,  Oi,  a-:,  .  .  .  &i,  &2  •  •  •  ,  are  calculated 
as  the  averages : 


ao  =  avg.  (?/)o^"; 
ai  =2  avg.  {y  cos  6)q~  ; 
a2  =  2  avg.  (?/  cos  26)0""; 
a„  =  2avg.  {y  cos  716)^^": 


2>T 


6i  =  2avg.  (11/sin  (9)o    ; 
62  =  2  avg.  (1/  sin  26)^'"; 
6„  =  2avg.  (?/sin  n^)o^"; 


(18) 


Hereby  any  individual  harmonic  can  be  calculated,  without 
calculating  the  preceding  harmonics. 

For  instance,  let  the  generator  c.m.f.  wave.  Fig.  44,  Tabic 
II,  column  2,  be  impressed  upon  an  underground  cable  system 


Fig.  44.     Generator  e.m.f.  wave. 


of  such  constants  (capacity  and  inductance),  that  the  natural 
frequency  of  the  system  is  G7()  cycles  per  second,  while  the 
generator  frequency  is  60  cycles.     The  natural  frequency  of  the 


TRIGONOMETRIC  SERIES. 


115 


circuit  is  then  close  to  that  of  the  11th  harmonic  of  the  generator 
wave,  660  cycles,  and  if  the  generator  voltage  contains  an 
appreciable  11th  harmonic,  trouble  may  result  from  a  resonance 
rise  of  voltage  of  this  frequency;  therefore,  the  11th  harmonic 
of  the  generator  wave  is  to  be  determined,  that  is,  an  and  6ii 
calculated,  but  the  other  harmonics  are  of  less  importance. 

Table  II 


e 

V 

cos  lie 

sin  115 

y  cos  115 

y  sin  we 

0 
10 
20 

5 

4 

20 

+  1.000 
-0.342 
-0.766 

0 
+  0.940 
-0.643 

+  5.0 

-1.4 

-15.3 

0 

+   3.8 
-12.9 

30 
40 
50 

22 
19 
25 

+  0.866 
+  0.174 
-0.985 

-0.500 
+  0.985 
-0.174 

+  19.1 

+  3.3 

-24.6 

-11.0 

+  18.7 
-   4.3 

60 
70 

80 

.       29 
29 
30 

+  0.500 
+  0.643 
-0.940 

-0.866 
+  0.766 
+  0.342 

+  14.5 
+  18.6 
-28.2 

-25.1 

+  22.2 
+  10.3 

90 

100 

110 

38 
46 
38 

0 
+  0.940 
-0.643 

- 1 . 000 
+  0.342 
+  0.766 

0 
+  43.3 
-24.4 

-38.0 
+  15.7 
+  29.2 

120 
130 
140 

41 
50 
32 

-0.500 
+  0.985 
-0.174 

-0.866 
-0  174 
+  0.985 

-20.5 

+  49.2 

-5.6 

-35.5 

-   8.7 
f  31.5 

150 
160 
170 

30 
33 

7 

-0.S66 
+  0.766 
+  0.342 

-0.500 
-0.643 
+  0.940 

-26.0 
+  25.3 

+  2.2 

-15.0 
-21.3 

180 

-5 

Divided 

Total.  .,  ...j^ 

+  34.5 
+  3.83  =  a„ 

-29.8 
-3.31=b„ 

In  the  third  column  of  Table  II  thus  are  given  the  values 
of  cos  11^,  in  the  fourth  column  sin  11^,  in  the  fifth  column 
y  cos  11^,  and  in  the  sixth  column  y  sin  11^.  The  former  gives 
as  average  +1.915,  hence  aii-=  +3.83,  and  the  latter  gives  as 
average  —1.655,  hence  6ii  =  — 3.31,  and  the  11th  harmonic  of 
the  generator  wave  is 

an  cos  11^  +&11  sin  11^  =  3.83  cos  11^-3.31  sin  11^ 
=  5.07  cos  (11^+410), 


116  ENGINEERING  MATHEMATICS. 

hence,  its  effective  value  is 

5.07 

—=-  =  3.58, 

\/2 

while  the  effective  value  of  the  total  generator  wave,  that 
is,  the  square  root  of  the  mean  squares  of  the  instanta- 
neous values  y,  is 

e  =  30.5, 

thus  the  11th  harmonic  is  11.8  per  cent  of  the  total  voltage, 
and  whether  such  a  harmonic  is  safe  or  not,  can  now  be  tleter- 
niined  from  the  circuit  constants,  more  particularly  its  resist- 
ance. 

82.  In  general,  the  successive  harmonics  decrease;  that  is, 
with  increasing  n,  the  values  of  a^  and  6,,  become  smaller,  and 
when  calculating  Un  and  6„  by  equation  (18),  for  higher  values 
of  n  they  are  derived  as  the  small  averages  of  a  number  of 
large  quantities,  and  the  calculation  then  becomes  incon- 
venient and  less  correct. 

Where  the  entire  series  of  coefficients  a„  and  6„  is  to  be 
calculated,  it  thus  is  preferable  not  to  use  the  complete  periodic 
function  y,  but  only  the  residual  left  after  subtracting  the 
harmonics  which  have  already  been  calculated;  that  is,  after 
tto  has  been  calculated,  it  is  subtracted  from  y,  and  the  differ- 
ence, ?/i  =?/— flo,  is  used  for  the  calculation  of  ai  and  hi. 

Then  Oi  cos  ^+?)i  sin  ^  is  subtracted  from  yi,  and  the 
difference, 

2/2  =  ?/i—  (fli  COS  6  +h]  sin  0) 
=  ?/—  (ao  +  tti  cos  ^  +  61  sin  d), 

is  used  for  the  calculation  of  aa  and  62- 

Then  a2  cos  2^+62  sin  26  is  subtracted  from  7/2,  and  the  rest, 
ys,  used  for  the  calculation  of  a-.i  and  63,  etc. 

In  this  manner  a  higher  accuracy  is  derived,  and  the  calcu- 
lation simplified  by  having  the  instantaneous  values  of  the 
function  of  the  same  magnitude  as  the  coefficients  a„  and  6„. 

As  illustration,  is  given  in  Table  III  the  calculation  of  the 
first  three  harmonics  of  the  pulsating  current,  Fig.  41,  Table  I: 


TRIGONOMETRIC  SERIES.  117 

83.  In  electrical  engineering,  the  most  important  periodic 
functions  are  the  alternating  currents  and  voltages.  Due  to 
the  constructive  features  of  alternating-current  generators, 
alternating  voltages  and  currents  are  almost  always  symmet- 
rical waves;  that  is,  the  periodic  function  consists  of  alternate 
half-waves,  which  are  the  same  in  shape,  but  opposite  in  direc- 
tion, or  in  other  words,  the  instantaneous  values  from  180  deg. 
to  360  deg.  are  the  same  numerically,  but  opposite  in  sign, 
from  the  instantaneous  values  between  0  to  180  deg.,  and  each 
cycle  or  period  thus  consists  of  two  equal  but  opposite  half 
cycles,  as  shown  in  Fig.  44.  In  the  earher  days  of  electrical 
engineering,  the  frequency  has  for  this  reason  frequently  been 
expressed  by  the  number  of  half-waves  or  alternations. 

In  a  symmetrical  wave,  those  harmonics  which  produce  a 
difference  in  the  shape  of  the  positive  and  the  negative  half- 
wave,  cannot  exist;  that  is,  their  coefficients  a  and  6  must  be 
zero.  Only  those  harmonics  can  exist  in  which  an  increase  of 
the  angle  6  by  180  deg.,  or  tz,  reverses  the  sign  of  the  function. 
This  is  the  case  with  cos  nd  and  sin  nd,  if  n  is  an  odd  number. 
If,  howTver,  n  is  an  even  number,  an  increase  of  /?  by  ;r  increases 
the  angle  nd  by  2?:  or  a  multiple  thereof,  thus  leaves  cos  nd 
and  sin  n^  with  the  same  sign.  The  same  applies  to  a^.  There- 
fore, symmetrical  alternating  waves  comprise  only  the  odd 
harmonics,  but  do  not  contain  even  harmonics  or  a  constant 
term,  and  thus  are  represented  by 

y  =  a\  cos  O+as  cos  3(9  +  as  cos  50  +  .  .  . 
+&1  sin  ^+&3sin  3^+65sin  5^  + (19) 

When  calculating  the  coefficients  a„  and  bn  of  a  symmetrical 
wave  by  the  expression  (18),  it  is  sufficient  to  average  from  0 
to  tt;  that  is,  over  one  half-wave  only.  In  the  second  half-wave, 
cos  nd  and  sin  nd  have  the  opposite  sign  as  in  the  first  half-wave, 
if  n  is  an  odd  number,  and  since  y  also  has  the  opposite  sign 
in  the  second  half-wave,  y  cos  nd  and  y  sin  nd  in  the  second 
half-wave  traverses  again  the  same  values,  with  the  same  sign, 
as  in  the  first  half-wave,  and  their  average  thus  is  given  by 
averaging  over  one  half-wave  only. 

Therefore,  a  symmetrical  univalent  periodic  function,  as  an 


118 


ENGINEERING  MATHEMATICS. 


Table 


c\  =  a\  cosfl 

e 

u 

y^=v-aa 

y,  cos  6 

1/,  sin  6 

+  bi  sin  e 

ui=y<\—c^ 

0 

-60 

-111 

-111 

0 

-84 

-27 

10 

-49 

-100 

-98 

-17 

-85 

-15 

20 

-38 

-89 

-84 

-30 

-S3 

-6 

30 

-26 

-77 

-67 

-38 

-79 

+  2 

40 

-12 

-63 

-48 

-40 

-72 

9 

50 

0 

-51 

-33 

-39 

-63 

12 

60 

+  11 

-40 

-20 

-35 

-52 

12 

70 

27 

-24 

-8 

-23 

-40 

16 

80 

39 

-12 

-2 

-12 

-26 

14 

90 

50 

-1 

0 

-1 

-11 

10 

100 

61 

+  10 

-2 

+  10 

+  4 

6 

110 

71 

20 

-7 

+  19 

18 

+  2 

120 

81 

30 

-15 

+  26 

32 

-2 

130 

90 

39 

-25 

+  30 

45 

-6 

140 

99 

48 

-37 

+  31 

58 

-10 

150 

107 

56 

-49 

+  28 

67 

-11 

160 

114 

63 

-59 

+  22 

75 

-12 

170 

119 

68 

-67 

+  12 

81 

-13 

180 

122 

71 

-71 

0 

84 

-13 

190 

124 

73 

-72 

-13 

85 

-12 

200 

126 

75 

-71 

-26 

83 

-8 

210 

125 

74 

-64 

-37 

79 

-5 

220 

123 

72 

—  55 

-47 

72 

0 

230 

120 

69 

-44 

-'l^ 

63 

+  6 

240 

116 

65 

-32 

<^ 

52 

13 

250 

110 

59 

-20 

40 

19 

260 

100 

49 

-9 

-48 

26 

23 

270 

85 

34 

0 

-34 

11 

23 

280 

65 

+  14 

+  2 

-14 

-4 

18 

290 

35 

-16 

-5 

+  15 

-18 

+  2 

300 

+  17 

-34 

-17 

+  30 

-32 

_2 

310 

0 

-51 

-33 

+  39 

-45 

-6 

320 

-13 

-64 

(  -75. 

?       -49 

+  41 

-58 

-6 

330 

-26 

-65 

+  37 

-67 

-8 

340 

-38 

"-89 

-84 

+  30 

-75 

-14 

350 

-49 

-100 

-99 

+  17 

-81 

-19 

Total  .  . 

f  1826 

Total 

.-1520 

-204 

Total 

Divided 

Divided  by 

Divided  by  18 .  . . 

by  36  ... 

+  50.7  =  Oo 

IS 

-84.4  =  a, 

-11.3  =  6, 

TRIGONOMETRIC  SERIES. 


119 


III. 


1-2  COS  2d 

1/2  sin  29 

c.  =  a2  cos  2f 
+  62siu2tf 

U3  =  Vt—ci 

i/3  COS  .3« 

1/3  sin  39 

e 

-27 

-14 

—  5 

0 
—  5 
-4 

-15 
-12 

-7 

-12 
-3 
+  1 

-12 

-3 

0 

0 

-1 

+  1 

0 
10 
20 

+  1 
+  2 
-2 

+  2 

+  9 

+  12 

-1 

+  4 

11 

+  3 
+  5 
+  1 

0 
_2 
-1 

+  3 

+  4 
0 

30 
40 
50 

-6 
-12 
-13 

+  10 
+  10 

+  5 

13 
15 
16 

-1 

+  1 
_2 

+  1 
-1 

+  1 

0 
0 

+  2 

60 
70 
80 

-10 
-6 
-2 

0 
_2 
-1 

15 

12 

7 

—  5 
-6 

—  5 

0 
-3 
-4 

+  5 
+  5 
+  2 

90 
100 
110 

+  1 
+  1 
-2 

+  2 

+  6 

+  10 

+  1 

-4 

-11 

-3 
—  2 

+  1 

-3 
-2 

0 

0 
-1 

+  1 

120 
130 
140 

-5 

-9 

-12 

+  10 
+  8 
-4 

-13 
-15 

-16 

+  2 
+  3 

+  3 

0 
-1 
-3 

+  2 
+  3 
+  1 

150 
160 
170 

-13 

-11 

-6 

0 
-4 
-6 

-15 
-12 

-7 

+  2 

0 

-1 

_2 
0 
0 

0 

0 

-1 

180 
190 
200 

-2 

0 

-1 

-4 
0 

+  6 

-1 

+  4 
11 

-4 
-4 
—  5 

0 
-2 
-4 

-4 
-4 
_2 

210 
220 
230 

-6 
-15 
-22 

+  11 
+  12 

+  8 

1 
13 
15 
16      i 

0 

+  4 
+  7 

0 

+  4 
+  3 

0 

+  2 
+  6 

240 
250 
260 

-23 

-17 

_2 

0 
-6 

-1 

15 
12 

7 

+  8 
+  6 
—  5 

0 
-3 

+  4 

+  8 
+  5 
_2 

270 
280 
290 

+  1 
+  1 
-1 

+  2 
+  6 

+  6 

+  1 

-4 

-11 

-3 
_2 

+  5 

+  3 

+  2 
_2 

0 

+  1 
-4 

300 
310 
320 

-4 
-11 
-18 

+  7 
+  9 
+  6 

-13 
-15 
-16 

+  5 
+  1 
-3 

0 

0 

-3 

—  5 
-1 

+  1 

330 
340 
350 

-270 
-15.0  =  0., 

+  120 
+  6.7  =  6, 

Total 

Divided  by  18 

-33 

-1.8  =  03 

+  27 
+  1.5  =  63 

120  ENGINEERING  MATHEMATICS. 

alternating  voltage  and  current  usually  is,  can  be  represented 
by  tlie  expression, 

y  =  ai  cos  0  +a^  cos  3  ^+^5  cos  5  0  +a7  cos  7d  +  . .  . 
-\-bi  sin  ^+63  sin  3  d+h^  sin  5  d  +  b7  sin  7  6  +...;       (20) 

where, 

«!  =  2  avg.  (y  cos  /9)o";  &i  =2  avg.  (ij  sin  6)^'; 

as  =2  avg.  {y  cos  2(9) q^;  63=2  avg.  (;/  sin  S^)^'; 

a5  =  2  avg.  (y  cos  o^)^'';  65  =  2  avg.  (1/  sin  5^)o"; 

07  =  2  avg.  (y  cos  76)0";  67  =  2  avg.  f?/  sin  76)^". 


.  (21) 


84.  From  180  deg.  to  360  deg.,  the  even  harmonics  have 
the  same,  but  the  odd  harmonics  the  opposite  sign  as  from  0 
to  180  deg.  Therefore  adding  the  numerical  values  in  the 
range  from  180  deg.  to  360  deg.  to  those  in  the  range  from  0 
to  180  deg.,  the  odd  harmonics  cancel,  and  only  the  even  har- 
monics remain.  Inversely,  by  subtracting,  the  even  harmonics 
cancel,  and  the  odd  ones  remain. 

Hereby  the  odd  and  the  even  harmonics  can  be  separated. 
If  y  =  y(^o)  are  the  numerical  values  of  a  periodic  function 
from  0  to  180  deg.,  and  y'  =  ij{d+7z)  the  numerical  values  of 
the  same  function  from  180  deg.  to  360  deg., 

y2{d)  =  h\y{0)+yid-^r:)\,     ....    (22) 

is  a  periodic  function  containing  only  the  even  harmonics,  and 

yi{0)  =  h\y(0)-y{d+rr)\ (23) 

is  a  periodic  function  containing  only  the  odd  harmonics ;  that  is : 

yi{0)=ai  cos  O+as  cos  3^+  a^  cos  5^  +  .  .  . 

+  &1  sin  ^  +  63sin  3  ^+65sin  5^+ ...;      .     .     (24) 

y-zid)  =^0+^2  cos  26  +a4  cos  46  +  .  .  . 

+  62  sin  2^ +  64  sin  4(9  +  ...,       (25) 

and  the  complete  function  is 

y{d)=y,{6)+y2(6) (26) 


TRIGONOMETRIC  SERIES.  121 

By  this  method  it  is  convenient  to  determine  whether  even 
harmonics  are  present,  and  if  they  are  present,  to  separate 
them  from  the  odd  harmonics. 

Before  separating  the  even  harmonics  and  the  odd  har- 
monics, it  is  usually  convenient  to  separate  the  constant  term 
ao  from  the  periodic  function  y,  by  averaging  the  instantaneous 
values  of  ]j  from  0  to  360  deg.  The  average  then  gives  tto, 
and  subtracted  from  the  instantaneous  values  of  y,  gives 

yo{d)=y{d)-ao (27) 

as  the  instantaneous  values  of  the  alternating  component  of  the 
periodic  function;  that  is,  the  component  yo  contains  only  the 
trigonometric  functions,  but  not  the  constant  term,  ?/o  is 
^*hen  resolved  into  the  odd  series  y\,  and  the  even  series  1/2. 
85.  The  alternating  wave  ^0  consists  of  the  cosine  components : 

u{d)  =ai  cos  0 -\-a2  cos  20 -\-az  cos  3^  +  a4  cos  4^  +  .  .  .,    (28) 

and  the  sine  components : 

v{e)  =  hi  sin  ^  +  62  sin  2/? +  63  sin  3^+  64  sin  4^  +  .  ..;    (29) 

that  is, 

y^{d)=u{0)-\-vid) (30) 

The  cosine  functions  retain  the  same  sign  for  negative 
angles  {—0),  as  for  j)ositive  angles(  +  ^),  w^hile  the  sine  functions 
reverse  their  sign;  that  is, 

u{-0)==+u{d)     and     v{-0)  =  -v{0).       .     .     .     (31) 

Therefore,  if  the  values  of  ?/o  for  positive  and  for  negative 
angles  6  are  averaged,  the  sine  functions  cancel,  and  only  the 
cosine  functions  remain,  while  by  subtracting  the  values  of 
iJq  for  positive  antl  for  nc^gative  angles,  only  the  sine  functions 
remain;  that  is, 

yQ(e)+yo{-d)=2u{d); 

(32) 

yo{d)-yo{-d)=2v{0): 

hence,  the  cosine  terms  and  the  sine  terms  can  be  separated 
from  each  other  by  combining  the  instantaneous  values  of  2/0 
for  positive  angle  6  and  for  negative  angle  {—0),  thus: 

u{d)  =  h{y^{d)^-yo{~d)\,  1 

(33) 

v{d)  =  h\yoi(^)-yo{-0)\. 


122  ENGINEERING  MATHEMATICS. 

Usually,  before  separating  the  cosine  ami  the  sine  terms, 
u  and  V,  first  the  constant  term  Qq  is  separated,  as  discussed 
above;  that  is,  the  alternating  function  VQ^y—aQ  used.  If 
the  general  periodic  function  y  is  used  in  equation  (33),  the 
constant  term  Qq  of  this  periodic  function  appears  in  the  cosine 
term  u,  thus: 

u{0)  =  \{y{0)  +y{-0)\  =ao+ai  cos  ^  +  a2  cos  2^ +  03  cos  3^  +  .  .  ., 

while  v{d)  remains  the  same  as  when  using  y^. 

86.  Before  separating  the  alternating  function  ?/o  into  the 
cosine  function  u  and  the  sine  function  v,  it  usually  is  more 
convenient  to  resolve  the  alternating  function  ?/q  into  the  odd 
series  t/i,  and  the  even  series  ?/2,  as  discussed  in  the  preceding 
paragraph,  and  then  to  separate  yi  and  1/2  each  into  the  cosine 
and  the  sine  terms: 

ui{6)-=^{yi(d)+yi{-d)\-=aiCOsd+a3CosSd-\-a5C.os5d+.  .  .;] 

(34) 
vi(d)  =  h\yi(d)-yi(-d)\=^his\nd+bssmSd+h5sm5d+..  .:J 

U2(0)  =  h{y2(d)  +y2i- 0)\  =a2Cos  26  +a^cos  46  + .  .  :  1 

.    .     (35) 
Void)  =  h\y2(d)-y2{- 6)\  =h2  sm  26  +  h4  sm  id  + .  .  .    J 

In  the  odd  functions  u\  and  Vi,  a  change  from  the  negative 
angle  (—6)  to  the  supplementary  angle  {rz—d)  changes  the  angle 
of  the  trigonometric  function  by  an  odd  multiple  of  tt  or  180 
deg.,  that  is,  by  a  multiple  of  2~  or  360  deg.,  plus  180  deg., 
which  signifies  a  reversal  of  the  function,  thus : 

Ux{6)=^h{yr(d)-y,{r:-6)\,  ] 

....      (36) 

vM  =  h\yii0)+yii--o)\.  J 

However,  in  the  even  functions  ?/2  and  V2  a  change  from  the 
negative  angle  (—^)  to  the  supplementary  angle  (7:—^),  changes 
the  angles  of  the  trigonometric  function  by  an  even  multiple 
of  7c ;  that  is,  by  a  multiple  of  2?:  or  360  deg.;  hence  leaves 
the  sign  of  the  trigonometric  function  unchanged,  thus : 

u2{6)  =  h\y2{d)+y2{7:-6)\,] 

....    (37) 
V2(d)  =  h{y2(0)-y2(7:-6)\.  J 


TRIGONOMETRIC  SERIES. 


123 


To  avoid  the  possibility  of  a  mistake,  it  is  preferable  to  use 
the  relations  (34)  and  (35^,  which  are  the  same  for  the  odd  and 
for  the  even  series. 

87.  Obviously,  in  the  calculation  of  the  constants  a„  and 
h„,  instead  of  averaging  from  0  to  180  deg.,  the  average  can 
be  made  from  —90  deg.  to  +90  deg.  In  the  cosine  function 
u{d),  however,  the  same  numerical  values  repeated  with  the 
same  signs,  from  0  to  —90  deg.,  as  from  0  to  +90  deg.,  and 
the  multipliers  cos  nO  also  have  the  same  signs  and  the  same 
numerical  values  from  0  to  —90  deg.,  as  from  0  to  +90  deg. 
In  the  sine  function,  the  same  numerical  values  repeat  from  0 
to  —90  deg.,  as  from  0  to  +90  deg.,  but  with  reversed  signs, 
and  the  multipliers  sin  nd  also  have  the  same  numerical  values, 
but  with  reversed  sign,  from  0  to  —90  deg.,  as  from  0  to  +90 
deg.  The  products  u  cos  nO  and  v  sin  nd  thus  traverse  the 
same  numerical  values  with  the  same  signs,  between  0  and 
—  90  deg.,  as  between  0  and    +90  deg.,  and  for  deriving  the 

averages,  it  thus  is  sufficient  to  average  only  from  0  to  — ,  or 

90  deg.;   that  is,  over  one  quandrant. 

Therefore,  by  resolving  the  periodic  function  y  into  the 
cosine  components  u  and  the  sine  components  v,  the  calculation 
of  the  constants  a„  and  &„  is  greatly  simplified;  that  is,  instead 
of  averaging  over  one  entire  period,  or  360  deg.,  it  is  necessary 
to  average  over  only  90  deg.,  thus: 

IT  r 

ai=2  avg.  (ui  cos  /9)o-  ;       i!>i  =  2  avg.  {vi  sin  ^)o-  ; 

n  r 

a2  =  2  avg.  (u2  cos  2(?)o- ;     62  =  2  avg.  {v2  sin  2(9)o-; 

aa  =  2  avg.  {u^  cos  3^)o'-  ;     &3  =  2  avg.  {vz  sin  3^)o^  ;  j. .       (38) 

a4  =  2  avg.  (m4  cos  4/9)o-  ;     64  =  2  avg.  (^4  sin  4^)0^  ; 

as  =  2  avg.  {us  cos  5(9)o2  ;     ?>5  =  2  avg.  (^5  sin  5^)o^; 
etc.  etc. 

where  Ui  is  the  cosine  term  of  the  odd  function  yi;  U2  the 
cosine  term  of  the  even  function  y2',  U3  is  the  cosine  term  of 
the  odd  function,  after  subtracting  the  term  with  cos  6]  that  is, 

U3  =  ui  —  ai  cos  0, 


124  ENGINEERING  MATHEMATICS. 

analogously,  U4  is  the  cosine  term  of  the  even  function,  after 
subtracting  the  term  cos  26; 

1(4  =  112— ao  cos  26, 

and  in  the  same  manner, 

1^5  =  ^3  —  03  cos  3^, 
■UG  =  ii4—a4  cos  40, 

and  so  forth;    Vi,  V2,  v-j,  V4,  etc.,  are  the  corresponding  sine 
terms. 

When  calculating  the  coefficients  a„  and  6„  by  averaging  over 
90  deg.,  or  over  180  (leg.  or  300  deg.,  it  must  be  kept  in  mind 
that  the  terminal  values  of  y  respectively  of  u  or  v,  that  is, 
the  values  for  ^  =  0  and  6  =  90  deg.  (or  0  =  180  deg.  or  360 
deg.  respectively)  are  to  be  taken  as  one-half  only,  since  they 
are  the  ends  of  the  measured  area  of  the  curves  a„  cos  n6  and 
hn  sin  n6,  which  area  gives  as  twice  its  average  height  the  values 
a„  and  6„,  as  discussed  in  the  preceding. 

In  resolving  an  empirical  periodic  function  into  a  trigono- 
metric series,  just  as  in  most  engineering  calculations,  the 
most  important  part  is  to  arrange  the  work  so  as  to  derive  the 
results  expeditiously  and  rapidly,  and  at  the  same  time 
accurately.  By  proceeding,  for  instance,  immediately  by  the 
general  method,  equations  (17)  and  (18),  the  work  becomes  so 
extensive  as  to  be  a  serious  waste  of  time,  while  by  the  system- 
atic resolution  into  simpler  functions  the  work  can  be  gi'eatly 
reduced. 

88.  In  resolving  a  general  periodic  function  y{6)  into  a 
trigonometric  series,  the  most  convenient  arrangement  is: 

1.  To  separate  the  constant  term  Qq,  by  averaging  all  the 
instantaneous  values  of  y(0)  from  0  to  360  deg.  (counting  the 
end  values  at  6  =  0  and  at  0  =  360  de<r.  one  half,  as  discussed 
above) : 

ao  =  avg.  \y(6)\o-, (10) 

and  then  subtracting  Oq  from  y{0),  gives  the  alternating  func- 
tion, 


TRIGONOMETRIC  SERIES. 


125 


2.  To   resolve   the   general  alternating  function   yQ(d)   into 
the  odd  function  y\{d),  and  the  even  function  y2iQ), 

ym  =  h\y^(0)-yo{0+^)\;     ....    (23) 
2/2(^)  =  Ji?yo(^)+?/o(^+^)i (22) 

3.  To  resolve  y\{0)  gnd  y2{0))  into  the  cosine  terms  ^^  and 
the  sine  terms  v, 

wi(^)  =  i!2/iW+!/,(-^)!:| 

U2(.d)  =  h\y2ie)+y2(-d)\;] 
V2{d)^h{y2(0)-y2i-0)\.\'    ' 

4.  To  calculate  the  constants  ai,  ao,  as.  .  .;  61,  ho,  63. 
by  the  averages, 


(34) 
(35) 


an  =  2  avg.  (unCos  n^JQ-' ; 


^    \.       ...     (38) 
6„  =  2  avg.  (v„  sin  nO)^-^ .  j 

If  the  periodic  function  is  known  to  contain  no  even  har- 
monics, that  is,  is  a  symmetrical  alternating  wave,  steps  1  and 
2  are  omitted. 


Fig.  45.     Mean  Daily  Temperature  at  Schenectady. 

89.  As  illustration  of  the  resolution  of  a  general  periodic 
wave  may  be  shown  the  resolution  of  the  observed  mean  daily 
temperatures  of  Schenectady  throughout  the  year,  as  shown 
in  Fig.  45.  up  to  the  7th  harmonic* 

*  The  numerical  values  of  temperature  cannot  claim  any  great  absolute 
accuracy,  as  they  are  averaged  over  a  relatively  small  number  of  years  only, 
and  observed  by  instruments  of  only  moderate  accuracy.  For  the  purpose 
of  illustrating  the  resolution  of  the  empirical  curve  into  a  trigonometric 
series,  this  is  not  essential,  however. 


126 


ENGINEERING  MA  THEM  A  TICS. 
Table  IV 


(1) 
e 

(2) 
V 

(3) 
y  —  aa  =  yo 

1/1 

(5) 

1/2 

Jan. 

0 
10 
20 

-  4.2 

-  4.7 

-  5.2 

-12.95 
-13.45 
-13.95 

-13.10 
-13.55 
-13.65 

+  0.15 
+  0.10 
-0.30 

Feb. 

30 
40 
50 

-  5.4 

-  3.8 

-  2.6 

-14.15 
-12.55 
-11.35 

-13.55 
-12.35 
-11.20 

-0.60 
-0.20 
-0.15 

Mar. 

60 
70 
80 

-    1.6 
+   0.2 

+    1.8 

-10.35 

-  8.55 

-  6.95 

-  9.75 

-  7.65 

-  6.05 

-0.60 
-0.90 
-0.90 

Apr. 

90 
100 
110 

+    5.1 
+   9.1 
+  11.5 

-   3.65 
+   0.35 
+   2.75 

-  3.35 

-  0.35 
+    1.75 

-0.30 
+  0.70 
+  1.00 

May 

120 
130 
140 

+  13.3 
+  15.2 

+  17.7 

+   4.55 
+   6.45 
+   8.95 

+   3.90 
+   5.85 
+   8.15 

+  0.65 
+  0.60 
+  0.80 

June 

150 
160 
170 

+  19.2 
+  19.5 
+  20.6 

+  10.45 
+  10.75 
+  11.85 

+  10.10 
+  10.80 
+  12.15 

+  0.35 
-0.05 
-0.30 

July 

ISO 
190 
200 

+  22.0 
+  22.4 
+  22.1 

+  13.25 
+  13.05 
+  13.35 

Aug. 

210 
220 
230 

+  21.7 
+  20.9 
+  19.8 

+  12.95 
+  12.15 
+  11.05 

Sept. 

240 
250 
260 

+  17.9 
+  15.5 
+  13.8 

+   9.15 
+   6.75 
+   5.15 

Oct. 

270 
280 
290 

+  11.8 
+   9.8 
+   8.0 

+   3.05 
+    1.05 
-   0.75 

Nov. 

300 
310 
320 

+    5.5 
+    3.5 
+    1.4 

-  3.25 

-  5.25 

-  7.35 

Dec. 

330 
340 
350 

-  1.0 

-  2.1 

-  3.7 

-   9.75 
-10.85 
-12.45 

Total 

Divided  by  36 . 

315.1 

8.75  =  ao 

1 

TRIGONOMETRIC  SERIES. 
Table  V. 


127 


(1) 
d 

(2) 
J/i 

(3) 

HI 

f4) 

(5) 

1/2 

(6) 

(7) 

-90 
-80 
-70 

+   3.35 
+   0.35 
-    1.75 

-0.30 
+  0.70 
+ 1 .  00 

-60 
-50 
-40 

-  3.90 

-  5.85 

-  8.15 

+  0.65 
+  0.60 
+  0  80 

-30 
-20 
-10 

0 
+  10 

+  20 

-10.10 
-10.80 
-12.15 

-13.10 
- 13 . 55 
-13.65 

+  0.35 
-0.05 
-0.30 

+  0.15 
+  0.10 
-0.30 

+  0.15 
-0.10 
-0.17 

0 
+  0.20 
-0.12 

-13.10 
-12.85 
-12.23 

0 
-0.70 
-1.42 

+  30 
+  40 
+  50 

-13.55 
-12.35 
-11.20 

-11.82 
-10.25 
-   8.53 

-1.73 
-2.10 
-2.67 

-0.60 
-0.20 
-0.15 

-0.12 
+  0.30 
+  0.22 

-0.47 
-0.50 
-0.37 

+  60 
+  70 
+  80 

-  9.75 

-  7.65 

-  6.05 

-  6.82 

-  4.70 

-  2.85 

-2.93 
-2.95 
-3.20 

-0.60 
-0.90 
-0.90 

+  0.02 
+  0.05 
-0.10 

-0.62 
-0.95 
-0.80 

+  90 

-    3.35 

0 

-3.35 

-0.30 

-0.30 

0 

i-' 

128 


ENGINEERING  MATHEMATICS. 


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129 


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ENGINEERING  MATHEMATICS. 


T.\BLE   VIII. 
COSINE   SERIES   w. 


(1) 
e 

(2) 

(3) 
w  cos  29 

(4) 
02  cos  26 

(5) 

(6) 
Ui  cos  49 

(7) 
(14COS49 

(8) 

1 
u«  COS  69 

0 
10 
20 

30 
40 
50 

60 
70 
80 
90 

+  0.15 
-0.10 
-0.17 

-0.12 
+  0.30 
+  0.22 

+  0.02 
+  0.05 
-0.10 
-0.30 

i(  +  0.15) 
-0.09 
-0.13 

-0.06 
+  0.05 
-0.04 

-0.01 

-0.04 

+  0.09 

K  +  0.30) 

0 
'  0  "  ' 

+  0.15 
-0.10 
-0.17 

-0.12 
+  0.30 
+  0 .  22 

+  0.02 
+  0.05 
-0.10 
-0.30 

^(  +  0.15) 
-0,08 
-0.03 

+  0.06 
-0.29 
-0.21 

-0.01 

+  0.01 

-0.08 

K  +  0.30) 

-0.16 
-0.12 
-0.03 

+  0.08 
+  0.15 
+  0.15 

+  0.08 
-0.03 
-0.12 
-0.16 

+  0.31 
+  0.02 
-0.14 

-0.20 
+  0.15 
+  0.07 

-0.06 
+  0.08 
+  0.02 
-0.14 

^(  +  0.31) 
"   +0.01 
+  0.07 

+  0.20 
-0.07 
+  0.03 

-0.06 
+  0.04 
-0.01 

i(  +  0.14) 

Total 

Divided  by 

9 

Multiplied 

by  2.... 

-0.01 

-0.001 

-0.002 

-0.71 

-0.079 

-0.158 
=  03 

+  0.44 

+  0.049 

+  0.098 
=  0, 

Table  IX. 

SINE   SERIES   V, 


(1) 
e 

(2) 

1'2 

(3) 
V2  sin  29 

(4) 
hi  sin  29 

(5) 

(6) 
VA  sin  49 

(7) 
hi  sin  49 

(8) 
vt 

(ft) 

vg  sin  69 

0 
10 
20 

30 
40 
50 

60 
70 
80 
90 

0 
+  0.20 
-0.12 

-0.47 
-0.50 
-0.37 

-0.62 

-0.95 

-0.80 

0 

0 
+  0.07 
-0.08 

-0.41 
-0.49 
-0.36 

-0.54 

-0.61 

-0.27 

0 

-0.20 
-0.39 

-0.52 
-0.59 
-0.59 

-0.52 
-0.39 
-0.20 

+  0.40 
+  0.27 

+  0.05 
+  0.09 
+  0.22 

-0.10 
-0.56 
-0.60 

+  0.26 
+  0.27 

+  0.04 
+  0.03 
-O.OS 

+  0.09 
+  0.55 
+  0.39 

+  0.22 
+  0.34 

+  0.30 
+  0.12 
-0.12 

-0.30 
-0.34 
+  0.22 

+  0.18 
-0.07 

-0.25 
-0.03 
+  0.34 

+  0.20 
-0.22 
-0.38 

+  0.16 

-0.07 

+  0 
+  0.03 
-0.30 

0 
-0.19 
-0.33 

Total 

Divided  bv  9 
Divided   by  2 

-2.69 

-0.30 

-0.60 

=  6, 

+  1.55 
+  0.172 
+  0.344 

-0.70 

-0.078 

-0.156 

TRIGONOMETRIC  SERIES.  131 

Table  IV  gives  the  resolution  of  the  periodic  temperature 
function  into  the  constant  term  ao,  the  odd  series  yi  and  the 
even  series  2/2. 

Table  V  gives  the  resolution  of  the  series  1/1  and  1/2  into 
the  cosine  and  sine  series  u\,  Vi,  U2,  V2. 

Tables  VI  to  IX  give  the  resolutions  of  the  series  mi,  Vi,  U2, 
V2,  and  thereby  the  calculation  of  the  constants  an  and  6„. 

90.  The  resolution  of  the  temperature  wave,  up  to  the 
7th  harmonic,  thus  gives  the  coefficients: 

ao=  +8.75; 

01  = -13.28;  61  =  -3.33; 

02= -0.001;  62= -0.602; 

03= -0.33;  63= -0.14; 

04= -0.154;  64= +0.386; 

05= +0.014;  65= -0.090; 

06= +0.100;  66  = -0.154; 

07= -0.022;  67  = -0.082; 
or,   transforming    by    the    binomial,    a„cosn^+6„sinn^  =  CnCOB 

(nd—fn),  by  substituting  Cn  =  N^^a^^  +  6„2  andtan?-„=—  gives, 

an 
ao=+8.75; 

ci  =  -13.69;     n  =  +  14.15°;  or  ri  =  +  14.15°; 
C2=-0.602;     ;'2=+89.r;°;     or  §=+44.95°+180n; 

C3=+0.359;  ;'3=-23.0°;     or  ^'=-7.7+120n=  +  112.3+120w; 

C4=-0.416;  r4=-68.2°;     or  ^^=-]7.05+90n=+72.95+90w; 

C5=+0.091;  r5=-81.15°;  or  ~  =  -]6.23+72n= +55.77+72m; 

C6=+0.184;  r6=-57.0°;     or  ^=-9.5+60n= +50.5+60m; 

C7=-0.085;     ^7=+7o.0°;    or  ^^=  +  10.7+51.4n, 
where  n  and  m  may  be  any  integer  number. 


132  ENGINEERING  MATHEMATICS. 

Since  to  an  angle  ;-„,  any  multiple  of  2-  or  360  (leg.  may 
be  added,  any  multiple  of  ' —  may  l)e  added  to  the  angle  — , 

and  thus  the  angle  —  may  be  made  positive,  etc. 

91.  The  equation  of  the  temperature  wave  thus  becomes: 

!/  =  8.75- 13.09  cos  (^- 14.15°) -0.(502  cos  2(^-44.95°) 
-0.359  cos  3(^-52.3") -0.416  cos  4(^-72.95°) 
-0.091  cos  5((9- 19.77°) -0.184  cos  6(^-20.5°) 
-0.085  cos  7(^-10.7°);  (a) 

or,  transformed  to  sine  functions  by  the  substitution, 
cos  w=— sin  (w  — 90°): 

i/  =  8.75 +  13.69  sin  (^-104.15°)  +0.602  sin  2(^-89.95°) 
+0.359  sin  3(i9-82.3°)  +0.416  sin  4(^-95.45°) 
+0.091  sin  5(^-109.77°)  +0.184  sin  6(^-95.5°) 
+  0.085  sin  7(^-75°).  (6) 

The  cosine  foi*m  is  more  convenient  for  some  purposes, 
the  sine  form  for  other  purposes. 

Substituting  ^5  =  ^-14.15°;  or,  a  =  ^- 104.15°,  these  two 
equations  (a)  and  (6)  can  be  transformed  into  the  form, 

?/  =  8.75- 13.69  cos  ./?-0.62  cos 209-30.8°) -0.359  cos3(/?-38.15°) 

-0.416  cos  4(.9- 58.8°) -0.091  cos  509-5.6°) 

-0.184  cos  6(/9- 6.35°) -0.085  cos  709-48.0°),  (c) 

and 

y-8.75+13.69 sin  oN  0.602  sin  2(a+14.2°)  +  0.359 sin  3(^+21.85°) 

+0.416  sin  4(0+8.7°)  +0.91  sin  5(o-5.6°) 

+0.184  sin  6(0^+8.65°)  +0.085  sin  7(o  +29.15°).  (d) 

The  periodic  variation  of  the  temperature  ?/,  as  expressed 
by  these  equations,  is  a  result  of  the  periodic  variation  of  the 
thermomotive  force;   that  is,  the  solar  radiation.     This  latter 


TRIGONOMETRIC  SERIES.  133 

is  a  minimum  on  Dec.  22d,  that  is,  9  tinio-dogrces  before  the 
zei'o  of  0,  hence  may  be  expressed  approximately  by: 

z  =  c-h  cos  {d  +  9°); 
or  substituting  /?  respectively  d  ior  6: 

z  =  c-h  cos  (/?+23.15°) 
.    =c+/isin  (a +  23.15°). 

This  means:  tlie  maximum  of  y  occurs  23.15  deg.  after  the 
naximum  of  z]  in  other  words,  the  temperature  lags  23.15  deg., 
or  about  ^  period,  b(^hind  the  thermomotive  force. 

Near  ^  =  0,  all  the  sine  functions  in  (d)  are  increasing;  that 
is,  the  temperature  wave  rises  steeply  in  spring. 

Near  ^  =  180  deg.,  the  sine  functions  of  the  odd  angles  are 
decreasing,  of  the  even  angles  increasing,  and  the  decrease  of 
the  temperature  wave  in  fall  thus  is  smaller  than  the  increase 
in  spring. 

The  fundamental  wave  greatly  preponderates,  with  ampli- 
tude ci  =  13.69. 

In  spring,  for  o  =  — 14.5  deg.,  all  the  higher  harmonics 
rise  in  the  same  direction,  and  give  the  sum  1.74,  or  12.7 
per  cent  of  the  fundamental.  In  fall,  for  o  =  — 14.5+;r,  the 
even  harmonics  decrease,  the  odd  harmonics  increase  the 
steepness,  and  give  the  sum   —0.67,  or   —4.9  per  cent. 

Therefore,  in  spring,  the  temperature  rises  12.7  per  cent 
faster,  and  in  autumn  it  falls  4.9  per  cent  slower  than  corre- 
sponds to  a  sine  wave,  and  the  difl'erence  in  the  rate  of  tempera- 
ture rise  in  spring,  antl  temperature  fall  in  autumn  thus  is 
12.7+4.9=17.6  per  cent. 

The  maximum  rate  of  temperature  rise  is  90  —  14.5  =  75.5 
deg.  behind  the  temperature  minimum,  and  23.15+75.5  =  98.7 
deg.  behind  the  minimum  of  the  thermomotive  force. 

As  most  periodic  functions  met  by  the  electrical  engineer 
are  symmetrical  alternating  functions,  that  is,  contain  only 
the  odd  harmonics,  in  general  the  work  of  resolution  into  a 
trigonometric  series  is  very  much  less  than  in  above  example. 
Where  such  reduction  has  to  be  carried  out  frequently,  it  is 
advisable  to  memorize  the  trigonometric  functions,  from  10 
to  10  deg.,  up  to  3  decimals;  that  is,  within  the  accuracy  of 
the  slide  rule,  as  thereby  the  necessity  of  looking  up  tables  is 


134  ENGINEERING  MATHEMATICS. 

eliminated  and  the  work  therefore  done  much  more  expe- 
ditiously. In  general,  the  slide  rule  can  be  used  for  the  calcula- 
tions. 

As  an  example  of  the  simpler  reduction  of  a  symmetrical 
alternating  wave,  the  reader  may  resolve  into  its  harmonics, 
up  to  the  7th,  the  exciting  current  of  the  transformer,  of  which 
the  numerical  values  are  given,  from  10  to  10  deg.  in  Table  X. 

C.  REDUCTION    OF    TRIGONOMETRIC    SERIES    BY  POLY- 
PHASE   RELATION. 

92.  In  some  cases  the  reduction  of  a  general  periodic  func- 
tion, as  a  complex  w^ave,  into  harmonics  can  be  carried  out 
in  a  much  quicker  manner  by  the  use  of  the  polyphase  equation, 
Chapter  III,  Part  A  (23).  Especially  is  this  true  if  the  com- 
plete equation  of  the  trigonometric  series,  which  represents  the 
periodic  function,  is  not  required,  but  the  existence  and  the 
amount  of  certain  harmonics  are  to  be  determined,  as  for 
instance  whether  the  periodic  function  contain  even  harmonics 
or  third  harmonics,  and  how  large  they  may  be. 

This  method  docs  not  give  the  coefficients  a„,  6„  of  the 
individual  harmonics,  but  derives  from  the  numerical  values 
of  the  general  wave  the  numerical  values  of  any  desired 
harmonic.  This  harmonic,  however,  is  given  together  with  all 
its  multiples;  that  is,  when  separating  the  third  harmonic, 
in  it  appears  also  the  6th,  9th,  12th,  etc. 

In  separating  the  even  harmonics  7/2  from  the  general 
wave  y,  in  paragraph  84,  by  taking  the  average  of  the  values 
of  y  for  angle  6,  and  the  values  of  y  for  angles  {d+iz),  this 
method  has  already  been  used. 

Assume  that  to  an  angle   0  there  is  successively  added  a 

constant  quantity  a,  thus:    6;    d  +  a;    6  + 2a;    d  +  Sa;    d  +  'ia, 

etc.,  until  the  same  angle  d  plus  a  multiple  of  2z  is  reached; 

2niT: 
d  +  na  =  d  +  2m7z;    that  is,   a  = ;    or,  in  other  words,  a  is 

1/n  of  a  multiple  of  2-.  Then  the  sum  of  the  cosine  as  well 
as  the  sine  functions  of  all  these  angles  is  zero: 

cos  (9-Kcos  (^  +  a)+cos  (^  +  2a)+cos  {d  +  Sa)+.  .  . 

H-eos  (^  +  [n-l]a)=0;       (1) 


TRIGONOMETRIC  SERIES.  135 

sin  ^+sin  ((9+a)+sin  {d +2a)  +sin  (d+3a)+.  .  . 

+sm(d+[n-l]a)=0,         (2) 

where 

na  =  2m- (3) 

These  equations  (1)  and  (2)  hold  for  all  values  of  a,  except  for 
a  =2tc,  or  a  multiple  thereof.  For  a  =  2;r  ob\dously  all  the  terms 
of  equation  (1)  or  (2)  become  equal,  and  the  sums  become 
n  cos  d  respectively  n  sin  6. 

Thus,  if  the  series  of  numerical  values  of  y  is  divided  into 

27Z- 

n  successive  sections,  each  covering  —  degrees,  and  these 
sections  added  together, 

y(o)  +y[^+'i)  +?/(^+2^)  +y{^+^^-t)  +•  •  • 

+  y(^0  +  [n-lT^y        (4) 

In  this  sum,  all  the  harmonics  of  the  wave  y  cancel  by  equations 
(1)  and  (2),  except  the  nth  harmonic  and  its  multiples, 

an  cos  nO+bn  sin  nO;   a2n  cos  2nO+b2,i  sin  2nd,  etc. 

in  the  latter  all  the  terms  of  the  sum  (4)  are  equal;  that  is, 
the  sum  (4)  equals  n  times  the  nth  harmonic,  and  its  multipk^s. 
Therefore,  the  nth  harmonic  of  the  periodic  function  y,  together 
with  its  multiples,  is  given  by 


yM=l 


For  instance,  for  n  =  2, 

y2  =  h\yiO)+y{0 +  ::)], 

gives  the  sum  of  all  the  even  harmonics;  that  is,  gives  the 
second  harmonic  together  with  its  multij^les,  the  4th,  6th,  etc., 
as  seen  in  paragraph  7,  and  for,  n  =  3. 


13G 


ENGINEERING  MATHEMATICS. 


gives  the  third  harmonic,  together  with  its  multiples,  the  6th, 
9th,  etc. 

This  method  does  not  give  the  mathematical  expression 
of  the  harmonics,  but  their  numerical  values.  Thus,  if  the 
mathematical  expressions  are  required,  each  of  the  component 
harmonics  has  to  be  reduced  from  its  numerical  values  to 
the  mathematical  equation,  and  the  method  then  usually  offers 
no  advantage. 

It  is  especially  suitable,  however,  where  certain  classes  of 
harmonics  are  desired,  as  the  third  together  with  its  multiples. 
In  this  case  from  the  numerical  values  the  effective  value, 
that  is,  the  equivalent  sine  wave  may  be  calculated. 

93.  As  illustration  may  be  investigated  the  separation  of 
the  third  harmonics  from  the  exciting  current  of  a  transformer. 

Table  X 


A 

(1) 
e 

(2) 
i 

(3) 

e 

(4) 
i 

(5) 

e 

(6) 
i 

(7) 

13 

0 
10 
20 

30 
40 
50 

60 

+  24.0 
+  20.0 
+  12 

+  4 

-  1.5 

-  6.5 

-  8.5 

120 
130 
140 

150 
160 
170 

180 

-15.1 
- 16 . 5 
-18.5 

-21 

-22.7 

-23.7 

-24 

240 
250 
260 

270 
280 
290 

300 

+  8.5 
+  10 
+  11 

+  12 
+  13 
+  14 

+  15.1 

+  5.8 
+  4.5 
+  1.5 

-1.7 
-3.7 
-5.4 

-5.8 

B 

35 

w 

3fl 

ia 

3d 

t3 

t9 

0 
30 
60 

+  5.8 
+  4.5 
+  1.5 

120 
150 
180 

-3.7 
-5.4 

-5.8  1 

240 
270 
300 

-1.5 

+  1.7 
+  3.7 

+  0.2 
+  0.3 
-0.2 

In  table  X  A,  are  given,  in  columns  1,  3,  5,  the  angles  6, 
from  10  deg.  to  10  deg.,  and  in  colunms  2,  4,  G,  the  correspond- 
ing values  of  the  exciting  current  i,  as  derived  by  calculation 
from  the  hysteresis  cycle  of  the  iron,  or  by  measuring  from  the 


TRIGONOMETRIC  SERIES. 


13- 


photographic  film  of  the  oscillograpli.  Column  7  then  gives 
one-third  the  sum  of  columns  2,  4,  and  (J,  that  is,  the  third  har- 
monic with  its  overtones,  is. 

To  find  the  9th  harmonic  and  its  overtones  ig,  the  same 
method  is  now  applied  to  is,  for  angle  3^.  This  is  recorded 
in  Table  X  B. 

In  Fig.  46  are  plotted  the  total  exciting  current  i,  its  third 
harmonic  is,  and  the  9th  harmonic  ig. 

This  method  has  the  advantage  of  showing  the  limitation 
of  the  exactness  of  the  results  resulting  from  the  limited  num- 


PiG.  46. 


ber  of  numerical  values  of  i,  on  which  the  calculation  is  based. 
Thus,  in  the  example.  Table  X,  in  which  the  values  of  i  arc 
given  for  every  10  deg.,  values  of  the  third  harmonic  arc  derived 
for  every  30  deg.,  and  for  the  9th  harmonic  for  every  90  deg.; 
that  is,  for  the  latter,  only  two  points  per  half  wave  are  deter- 
minable from  the  numerical  data,  and  as  the  two  points  per  half 
wave  are  just  sufficient  to  locate  a  sine  wave,  it  follows  that 
within  the  accuracy  of  the  given  numerical  values  of  i,  the 
9th  harmonic  is  a  sine  wave,  or  in  other  words,  to  determine 
whether  still  higher  harmonics  tlian  the  9th  exist,  requires  for 
i  more  numerical  values  than  for  every  10  deg. 

As  further  practice,  the  reader  may  separate  from  the  gen- 


138 


ENGINEERING  MA  THEM  A  TICS. 


eral  wave  of  current,  i'o  in  Tabic  XI,  the  even  harmonics  22, 
by  above  method, 

i2  =  Mio(0)  +io(0+lSO  deg.)], 

and  also  the  sum  of  the  odd  harmonics,  as  the  residue, 


ti=to  — 12, 

then  from  the  odd  harmonics  ii  may  be  separated  the  third 
harmonic  and  its  multiples, 

i3  =  HiiiO)  +ti(^  +  120  deg.  )  +ii(^+240  deg.)}, 

and  in  the  same  manner  from  is  may  be  separated  its  third 
harmonic ;  that  is,  ig. 

Furthermore,  in  the  sum  of  even  harmonics,  from  12  may 
again  be  separated  its  second  harmonic,  u,  and  its  multiples, 
and  therefrom,  ig,  and  its  third  harmonic,  ie,  and  its  multiples, 
thus  giving  all  the  harmonics  up  to  the  9th,  with  the  exception 
of  the  5th  and  the  7th.  These  latter  two  would  require  plotting 
the  curve  and  taking  numerical  values  at  different  intervals, 
so  as  to  have  a  number  of  numerical  values  di^^sible  by  5  or  7. 

It  is  further  recommended  to  resolve  this  unsymmetrical 
exciting  current  of  Table  XI  into  the  trigonometric  series  by 
calculating  the  coefficients  a„  and  6„,  up  to  the  7th,  in  the  man- 
ner discussed  in  paragraphs  6  to  8. 

Table  XI 


e 

to 

e 

to 

B 

to 

e 

to 

0 
10 
20 

+  95.7 
+  78.7 
+  53.7 

90 
100 
110 

-26.7 
-27.3 

-28.1 

180 
190 
200 

-34.3 
-27.3 

-16.8 

270 
280 
290 

-  3.3 

-  1.8 
+  1.2 

30 
40 
50 

+  23.7 
-  2.3 
-16.3 

120 
130 
140 

-28.8 
-29.3 
-29.8 

210 
220 
230 

-11.3 

-  8.3 

-  7.3 

300 
310 
320 

+  4.7 
+  10.7 

+  22.7 

60 
70 
80 

-22.8 
-24.3 

-25.8 

150 

160 

!   170 

-31 

-32.6 

-33.8. 

240 
250 
260 

-  6.3 

-  5.3 

-  4  3 

330 
340 
350 

+  41.7 
+  65.7 
+  85.7 

TRIGONOMETRIC  SERIES.  139 

D.   CALCULATION    OF    TRIGONOMETRIC    SERIES    FROM 
OTHER  TRIGONOMETRIC    SERIES. 

94.  An  hydraulic  generating  station  has  for  a  long  time  been 
supplying  electric  energy  over  moderate  distances,  from  a  num- 
ber of  750-kw.  4400-volt  60-cycle  three-phase  generators.  The 
station  is  to  be  increased  in  size  by  the  installation  of  some 
larger  modern  three-phase  generators,  and  from  this  station 
6000  kw.  are  to  be  transmitted  over  a  long  distance  transmis- 
sion line  at  44,000  volts.  The  transmission  line  has  a  length 
of  60  miles,  and  consists  of  three  wires  No.  0  B.  &  S.  with  5 
ft.  between  the  wires. 

The  question  arises,  whether  during  times  of  light  load  the 
old  750-kw.  generators  can  be  used  economically  on  the  trans- 
mission line.  These  old  machines  give  an  electromotive  force 
wave,  which,  like  that  of  most  earlier  machines,  differs  con- 
siderably from  a  sine  wave,  and  it  is  to  be  investigated,  whether, 
due  to  this  wave-shape  distortion,  the  charging  current  of  the 
transmission  line  will  be  so  greatly  increased  over  the  value 
which  it  would  have  with  a  sine  wave  of  voltage,  as  to  make 
the  use  of  these  machines  on  the  transmission  line  uneconom- 
ical or  even  unsafe. 

Oscillogi'ams  of  these  machines,  resolved  into  a  trigonomet- 
ric series,  give  for  the  voltage  between  each  terminal  and  the 
neutral,  or  the  Y  voltage  of  the  three-phase  system,  the  equa- 
tion: 

e  =  eo|sin  0-0.12  sin  (30- 2. 3°) -0.23  sin  (5^-1.5°) 

+0.13  sin  (7^ -6. 2°) I.     .     (1) 

In  first  approximation,  the  line  capacity  may  be  considered 
as  a  condenser  shunted  across  the  middle  of  the  line ;  that  is, 
half  the  line  resistance  and  half  the  line  reactance  is  in  series 
with  the  line  capacity. 

As  the  receiving  apparatus  do  not  utilize  the  higher  har- 
monics of  the  generator  wave,  when  using  the  old  generators, 
their  voltage  has  to  be  transformed  up  so  as  to  give  the  first 
harmonic  or  fundamental  of  44,000  volts. 

44,000  volts  between  the  lines  (or  delta)  gives  44,000  ^  VS  = 
25,400  volts  between  line  and  neutral.     This  is  the  effective 


140  ENGINEERING  MATHEMATICS. 

value,  and  the  niaxiniuni  value  of  the  fundamental  voltage 
wave  thus  is:  25,400 X \/2  =  36,000  volts,  or  36  kv.;  that  is, 
Co  =  36,  and 

e  =  36{sin  ^-0.12  sin  (3i9-2.3°)-0.23  sin  (5^-1.5°) 

+  0.13  sin  (7^-0.2°)!,   .     (2) 

would  be  the  voltage  supplied  to  the  transmission  line  at  the 
high  potential  terminals  of  the  step-up  transformers. 

From  the  wire  tables,  the  resistance  per  mile  of  No.  0  B.  &  S. 
copper  line  wire  is  ro=0.52  ohm. 

The  inductance  per  mile  of  wire  is  given  by  the  formula ; 

Lo --=0.7415  log  ^  +  0.0805mh,     ....     (3) 

where  U  is  the  distance  between  the  wires,  and  Ij.  the  radius  of 
the  wire. 

In  the  present  case,  this  gives  /« =  5  ft.  =  00  in.  Z^  =  0 .  1625  in. 
L()  =  1.9655  mh.,  and,  herefrom  it  follows  that  the  reactance,  at 
/=  60  cycles  is 

Jo  =  27r/Lo  =  0 .  75  ohms  per  mile (4) 

The  capacity  i)or  mile  of  wire  is  given  by  the  formula: 

Cq  = r-mf.; (5) 

hence,  in  the  present  case,  Co  =0.0159  mf.,  and  the  condensive 
reactance  is  derived  herefrom  as : 

^«  =  rr^^r^  =  l^>^0<^<^ohins;        ....      (6) 

ItzJL  0 
60  miles  of  line  then  give  the  condensive  reactance, 

X 

^f  =  7Tf!  =  2770  ohms; 
60  ' 

30  miles,  or  half  the  line  (from  the  generating  station  to  the 
middle  of  the  line,  where  the  line  cajiacity  is  represented  by  a 
shunted  condenser)  give:   the  resistance,  r  =  30ro=  15.6  ohms; 


TRIGONOMETRIC  SERIES.  141 

the  inductive  reactance,  a:  =  30xo  =  22.5  ohms,  and  the  equiva- 
lent circuit  of  the  line  now  consists  of  the  resistance  r,  inductive 
reactance  x  and  condensive  reactance  Xc,  in  series  with  each 
other  in  the  circuit  of  the  supply  voltage  e. 

95.  If  i=  current  in  the  line  (charging  current)  the  voltage 
consumed  by  the  Une  resistance  r  is  ri. 

The  voltage  consumed  by  tlie  inductive  reactance  x  h  x-r-; 

the  voltage  consumed  l:>y  the  condensive  reactance  Xc  is  Xc  |  idO, 
and  therefore, 

e  =  x^i  +  ri  +  x,.  i  idO (7) 


dO 


Differentiating  this  ecjuation,  for  the  purpose  of  eliminating 
the  integral,  gives 

de       dH       di 

dO^^W-^^dll^^'^'' 
or  \.     ...     (8) 

The  voltage  e  is  given  l)y  (2),  which  (Mjuation,  by  resolving 
the  trigonometric  functions,  g'ves 

e  =  3()  sin  ^  -  4 .  32  sin  3^  -  S .  28  sin  5^  +4 .  G4  sin  16 

+0.18  cos  '^0  f  0 . 22  cos  5^ - 0 . 50  cos  Id)    .     (9) 

hence,  differentiating, 

de 

_  =  3r,  cos  ^-  12.9()  cos  3/?-41.4  cos  5^+32.5  cos  Id 
da 

-0.54  sin  3^^-1.1  sin  5^  +  3.5  sin  70.     .     (10) 

Assuming  now  for  the  current  i  a  tiigonometric  series  with 
indeterminate  coefficients, 

i  =  ai  cos  0+a:i  cos  ^O+a^  cos  5^ +07  cos  70 

-f  61  sin  O+h.i  sin  SO+h^  sin  5^  +^7  sin  70,      .     (11) 


142 


ENGINEERING  MATHEMATICS. 


substitution  of  (10)  aiitl  (11)  into  equation  (8)  must  give  an 
identity,  from  which  equations  for  the  determination  of  an  and 
bn  are  derived;  that  is,  since  the  product  of  substitution  must 
be  an  identity,  all  the  factors  of  cos  6,  sin  6,  cos  3^,  sin  Sd, 
etc.,  must  vanish,  and  this  gives  the  eight  equations: 


30       =2770ai+   15.66i-     22.5ai;l 
0         =27706i-    15.6ai-     22.56i; 

-12.9G  =  2770a3+   46.863-   202. Saa; 

-  0.54  =  277063-   46.8a3-   202.563: 

-41.4   =2770a5+       7865-   5G2.5a5; 

-1.1    =277065-       780,5-   56.2565; 

32.5   =2770a7 +  109.267- 1102.507; 

3.5   =277067-109.207-1102.567. 


(12) 


Resolved,  these  equations  give 

Oi=  13.12 
61=  0.07 
03  =  -  5.03; 
63=-  0.30: 
05= -18.72; 
65=-  1.15; 
07=  19.30: 
67=  3.37: 
hence, 

1  =  13 .  12  cos  ^  -  5 .  03  cos  3^  - 18 .  72  cos  50  + 19 .  30  cos  70 
+  0.07  sin  ^-0.30  sin  3^-1.15  sin  5^+3.37  sin  70 
=  13.12  cos  (^-0.3°)-5.04cos  (3^-3.3°) 
-18.76  cos  (5^-3.6°) +19.59  cos  (7^-9.9°). 


(13) 


(14) 


TRIGONOMETRIC  SERIES.  143 

96.  The  effective  value  of  this  current  is  given  as  the  square 
root  of  the  sum  of  squares  of  the  effective  values  of  the  indi- 
vidual harmonics,  thus : 


a/2|^  + 2^21.6  amp. 


As  the  voltage  between  line  and  neutral  is  25,400  effective, 
this  gives  Q  =  25,400X2 1.6  =  540, 000  volt-amperes,  or  540  kv.- 
amp.  per  line,  thus  a  total  of  3Q  =  1620  kv.-amp.  charging  cur- 
rent of  the  transmission  line,  when  using  the  c.m.f.  wave  of 
these  old  generators. 

It  thus  would  require  a  minimum  of  3  of  the  750- kw. 
generators  to  keep  the  voltage  on  the  line,  even  if  no  power 
whatever  is  delivered  from  the  line. 

If  the  supply  voltage  of  the  transmission  line  were  a  perfect 
sine  wave,  it  would,  at  44,000  volts  between  the  lines,  be  given 
by 

ei=36sin  ^,    .....     .     (15) 

which  is  the  fundamental,  or  first  harmonic,  of  equation  (9). 

Then  the  current  i  would  also  be  a  sine  wave,  and  would  be 
given  by 

ii  =  a\  cos  ^+61  sin  d, 
=  13.12  cos  ^+0.07  sin  ^,     -,     .     .     .     (16) 
=  13.12  cos  (^-0.3°), 

and  its  effective  value  would  be 

13  r^ 

/i=— ^  =  9.3amp.       .....     (17) 

This  would  correspond  to  a  kv.-amp.  input  to  the  line 

3Qi  =3X25.4X9.3  =  710  kv.-amp. 

The  distortion  of  the  voltage  w^ave,  as  given  by  equation  (1), 
thus  increases  the  charging  volt-amperes  of  the  line  from  710 


144 


ENGINEERING  MATHEMATICS. 


kv.-anip.  to  1620  kv.-amp.  or  2.28  times,  and  while  with  a  sine 
wave  of  voltage,  one  of  the  750- kw.  generators  would  easily  be 
able  to  supply  the  charging  current  of  the  line,  due  to  the 


Fig.  47. 


wave  shape  distortion,  more  than  two  generators  are  required. 
It  would,  therefore,  not  be  economical  to  use  these  generators 
on  the  transmission  Hne,  if  they  can  Ije  used  for  any  other 
purposes,  as  short-distance  distribution. 


Fig.  48. 


In  Figs.  47  and  48  are  plotted  the  voltage  wave  and  tb'^ 
current  wave,  from  equations  (9)  and  (14)   respectively,  and 


TRIGONOMETRIC  SERIES.  145 

the  numerical  values,  from  10  (l(>g.  to  10  cleg.,  recorded  in 
Table  XII. 

In  Figs.  47  and  48  the  fundamental  sine  wave  of  voltage 
and  current  are  also  shown.  As  seen,  the  distortion  of  current 
is  enormous,  and  the  higher  harmonics  predominate  over  the 
fundamental.  Such  waves  are  occasionally  observed  as  charg- 
ing currents  of  transmission  lines  or  cable  systems. 

97.  Assuming  now  that  a  reactive  coil  is  inserted  in  series 
with  the  transmission  line,  between  the  step-up  transformers 
and  the  line,  what  will  be  the  voltage  at  the  terminals  of  this 
reactive  coil,  with  the  distorted  wave  of  charging  current 
traversing  the  reactive  coil,  and  how  does  it  compare  with  the 
voltage  existing  with  a  sine  wave  of  charging  current? 

Let  L  =  inductance,  thus  x  =  2;r/L  =  reactance  of  the  coil, 
and  neglecting  its  resistance,  the  voltage  at  the  terminals  of 
the  reactive  coil  is  given  by 

^—4 (•«) 

Substituting  herein  the  equation  of  current,  (11),  gives  ■ 
e'-=x\ai  sin  0+Sas  sin  ^O+Sa^  sin  dd+la-j  sin  70  1 

—  61  cos  ^—363  cos3^— 5/>5  cos  5(9— 767  cos  70] ;    J 
hence,  substituting  the  numerical  values  (13), 
e'  =  xi  13.12  sin  ^-15.09  sin  3^-93.0  sin  5^ +135.1  sm  70  1 
-0.07  cos  e  +0.90  cos  3^+5.75  cos  5^-23.()  cos  7^ ! 
=  xj  13.12  sin  (^-0.3°) -15.12  sin  (3^-3.3°) 
-93.8  sin  (5^-3.G°)  +139.1  sin  (7^;-9.9°)  |. 
This  voltage  gives  the  effective  value 


(19) 


K20) 


E'  =  xVi-|13.122  +  15.l2^+93.8^  +  139.Pi=119.4.r, 

while  the  effective  value  with  a  sine  wave  would  be  from  (17), 

Ei'  =  xIi=Q.Sx; 

hence,  the  voltage  across  the  reactance  x  has  been  increased 
12.8  times  by  the  wave  distortion. 


146 


ENGINEERING  MA  THEM  A  TICS. 


The  instantaneous  values  of  the  voltage  e'  are  given  in  the 
last  column  of  Table  XII,  and  plotted  in  Fig.  49,  for  x  =  l. 
As  seen  from  Fig.  49,  the  fundamental  wave  has  practically 


Fig.  49. 


vanished,  and  the  voltage  wave  is  the  seventh  harmonic,  modi- 
fied by  the  fifth  harmonic. 


Table  XII 


e 

e 

I 

e' 

e 

e 

t 

^' 

0 
10 
20 

-0.10 

+  2.23 

3.74 

+  8.67 
+  5.30 
-  0.86 

-    17 
+  46 
+  3 

90 
100 
110 

27.41 
31.77 
40.57 

-  4.15 
+  26.19 
+  24.99 

-2G0 
-106 
+  119 

30 
40 
50 

7.47 
17.35 
31.70 

+  7.39 
+  30.39 

+  38.58 

+  131 
-116 
+  36 

120 
130 
140 

42.70 
33.14 
18.03 

-  8.10 
-38.79 
-36.65 

+  182 
+  93 
-  96 

60 
70 
80 

42.06 
40.33 
32.87 

+  15.66 
-19.01 
-29.13 

+  167 
+  159 
-  54 

150 
160 
170 

6.99 

2.88 
1.97 

-13.41 
+  2.43 
-  1.00 

- 138 
-  31 
+  54 

90 

27.41 

-  4.15 

-200 

180 

+  0.10 

-  8.67 

+  17 

CHAPTER  IV. 


MAXIMA   AND    MINIMA. 


98.  In  engineering  investigations  the  problem  of  determin- 
ing the  maxima  and  the  minima,  that  is,  the  extrema  of  a 
function,  frequently  occurs.  For  instance,  the  output  of  an 
electric  machine  is  to  be  found,  at  which  its  efficiency  is  a  max- 
imum, or,  it  is  desired  to  determine  that  load  on  an  induction 
motor  which  gives  the  highest  power-factor;    or,  that  voltage 


Y 

,/' 

^ 

\ 

1 

/ 

\. 

:/ 

"pT 

PN. 

p 

p 

J-f 

(q 

> 

V 

X 

.— i— 

^ 

y 

-^ 

7 

V 

p 

y 

y 

/ 

'4 

P. 

/ 

r 

/ 

/\0 

0 

/ 

X 

Fig.  50.     Graphic  Solution  of  Maxima  and  Minima. 

which  makes  the  cost  of  a  transmission  line  a  minimum;  or, 
that  speed  of  a  steam  turbine  which  gives  the  lowest  specific 
steam  consumption,  etc. 

The  maxima  and  minima  of  a  function,  y--=f{x),  can  be  found 
by  plotting  the  function  as  a  curve  and  taking  from  the  curve 
the  values  x,  y,  which  give  a  maximum  or  a  minimum.  For 
instance,  in  the  curve  Fig.  50,  maxima  are  at  Pi  and  P2,  minima 
at  P3  and  P4.  This  method  of  determining  the  extrema  of 
functions  is  necessary,  if  the  mathematical  expression  between 

147 


148 


ENGINEERING  MATHEMATICS. 


X  and  y,  that  is,  the  function  y=f{x),  is  unknown,  or  if  the 
function  y=f{x)  is  so  complicated,  as  to  make  the  mathematical 
calculation  of  the  extrema  impracticable.  As  examples  of 
this  method  the  following  may  be  chosen: 

B 


16 

, 

— ■ 

— 

14 

12 

/ 

/ 

10 

8- 

C 

/ 

•2. 

/ 

/ 

/ 

H 

^ 

Y 

}    \ 

}    1 

}   1 

i   1 

1       16      18      20      22      24       26      28      30 

Fig.  51.     Magnetization  Curv^e. 

Example  i.  Determine  that  magnetic  density  B,  at  which 
tl<e  permeabihty  ju  of  a  sample  of  iron  is  a  maximum.  The 
relation  between  magnetic  field  intensity  H,  magnetic  density 
B  and  permeability  u  cannot  be  expressed  in  a  mathematical 
e  juation,  and  is  therefore    usually  given  in  the  form  of  an 


1400 
i200 

^ 

W 

V 

' 

/^mc 

N 

\ 

-600 

y 

/ 

S 

k 

y 

/ 

\ 

\, 

A)0- 

/ 

\ 

B 

' 

1 

3 

)     ( 

\ 

i 

\ 

1     1 

^     1 

Kil 
1      1 

3-lines 
2      13      1 

4       1 

5 

Fig.  52.     Permeability  Curve. 

empirical  curve,  relating  B  and  U,  as  shown  in  Fig.  51.     From 
this  curve,  corresponding  values  of  B  antl  R  are  taken,  and  their 

ratio,  that  is,  the  permeability  y« =77,  plotted  against  fi as  abscissa. 

H 

This  is  done  in  Fig.  52.     Fig.  52  then  shows  that  a  maximum 


MAXIMA  AND  MINIMA. 


149 


occurs  at  point  /tmax*  for  -5=10.2  kilolines,  /t  =  1340,  and  minima 
at  the  starting-point  P2,  for  B=0,  /f  =  370,  and  also  for  B=oo  , 
where  by  extrapolation  /i  =  l. 

Example  2.  Find  that  output  of  an  induction  motor 
which  gives  the  highest  power-factor.  \Miile  theoretically 
an  equation  can  be  found  relating  output  and  power-factor 
of  an  induction  motor,  the  equation  is  too  complicated  for  use. 
The  most  convenient  way  of  calculating  induction  motors  is 
to  calculate  in  tabular  form  for  different  values  of  slip  s,  the 
torque,  output,  current,  power  and  volt -ampere  input,  efficiency, 
power-factor,  etc.,  as  is  explained  in  "Theoretical  Elements 
of  Electrical  Engineering,"  third  edition,  p.  3G3.     From  this 


Cos0 
0.90 

0.88 
0.86 


Fig.  53.     Power-factor  Maximum  of  Induction  Motor. 


P. 

Po 

4 — 

R 

/ 

^ 

^ 

N, 

/ 

\ 

\, 

/ 

/ 

\ 

s. 

\ 

2( 

30 

30 

DO 

40 

P 

00 

50 

00 

'6C 

90    V 

/atts 

table  corresponding  values  of  power  output  P  and  power- 
factor  cos  6  are  taken  and  plotted  in  a  curve.  Fig.  53,  and  the 
maxinmm  derived  from  this   curve   is  P  =  4120,  cos  ^  =  0.9.04. 

For  the  purpose  of  determining  the  maximum,  obviously 
not  the  entire  curve  needs  to  be  calculated,  but  only  a  short 
range  near  the  maximum.  This  is  located  by  trial.  Thus 
in  the  ])r(^sent  instance,  P  and  cos  d  are  calculated  for  s  =  0.1 
and  s  =  0.2.  As  the  latter  gives  lower  power-factor,  the  maximum 
power-factor  is  below  s  =  0.2.  Then  s  =  0.05  is  calculated  and  gives 
a  higher  value  of  cos  0  than  s  =  0.1;  that  is,  the  maximum  is 
below  s  =  0.1.  Then  s  =  0.02  is  calculated,  and  gives  a  lower 
value  of  cos  0  than  s  =  0.05.  The  maximum  value  of  cos  6 
thus  lies  between  s  =  0.02  and  s  =  0.1,  and  only  the  part  of  the 
curve  between  s  =  0.02  and  s  =  0.1  needs  to  be  calculated  for 
the  determination  of  the  maximum  of  cos  6,  as  is  done  in  Fig.  53. 

99.  When  determining  an  extremum  of  a  function  y=f(x). 
by  plotting  it  as  a  curve,  the  value  of  x,  at  which  the  extreme 


150  ENGINEERING  MATHEMATICS. 

occurs,  is  more  or  less  inaccurate,  since  at  the  extreme  the 
curve  is  horizontal.  For  instance,  in  Fig.  53,  the  maximum 
of  the  curve  is  so  fiat  that  the  value  of  power  P,  for  which 
cos  0  became  a  maximum,  may  be  anywhere  between  P  =  4000 
and  P  =  4300,  within  the  accuracy  of  the  curve. 

In  such  a  case,  a  higher  accuracy  can  frequently  be  reached 
by  not  attempting  to  locate  the  exact  extreme,  but  two  points 
cf  the  same  ordinate,  on  each  side  of  the  extreme.  Thus  in 
Fig.  53  the  power  Pq,  at  which  the  maximum  power  factor 
cos  ^^  =  0.904  is  reached,  is  somewhat  uncertain.  The  value  of 
power-factor,  somewhat  below  the  maximum,  cos  ^  =  0.90, 
is  reached  before  the  maximum,  at  Pi  =  3400,  and  after  the 
maximum,  at  P2  =  4840.  The  maximum  then  may  be  calculated 
as  half-way  between  Pi  and  P2,  that  is,  at  Po  =  h\P\+P2\  = 
4120  watts. 

This  method  gives  usually  more  accurate  results,  but  is 
based  on  the  assumption  that  the  curve  is  synmietrical  on 
both  sides  of  the  extreme,  that  is,  falls  off  from  the  extreme 
value  at  the  same  rate  for  lower  as  for  higher  values  of  the 
abscissas.  \\Tiere  this  is  not  the  case,  this  method  of  inter- 
polation does  not  give  the  exact  maximum. 

Example  3.  The  efficiency  of  a  steam  turbine  nozzle, 
that  is,  the  ratio  of  the  kinetic  energy  of  the  steam  jet  to  tlu^ 
energy  of  the  steam  available  between  the  two  pressures  between 
which  the  nozzle  operates,  is  given  in  Fig.  54,  as  determined  by 
experiment.  As  abscissas  are  used  the  nozzle  mouth  opening, 
that  is,  the  widest  part  of  the  nozzle  at  the  exhaust  end,  as 
fraction  of  that  corresponding  to  the  exhaust  pressure,  while 
the  nozzle  throat,  that  is,  the  narrowest  part  of  the  nozzle,  is 
assumed  as  constant.  As  ordinatcs  are  plotted  the  efficiencies. 
This  curve  is  not  symmetrical,  but  falls  off  from  the  maximum, 
on  the  sides  of  larger  nozzle  mouth,  far  more  rapidly  than  on 
the  side  of  smaller  nozzle  mouth.  The  reason  is  that  with 
too  large  a  nozzle  mouth  the  expansion  in  the  nozzle  is  carried 
below  the  exhaust  pressure  7)2,  and  steam  eddies  are  produced 
by  this  overexpansion. 

The  maximum  efficiency  of  94.6  per  cent  is  found  at  the  point 
Po,  at  which  the  nozzle  mouth  corresponds  to  the  exhaust 
pressure.  If,  however,  the  maximum  is  determined  as  mid- 
way between  two  points  Pi  and  P2,  on  each  side  of  the  maximum, 


MAXIMA  AND  MINIMA. 


151 


at  which  the  efficiency  is  the  same,  93  per  cent,  a  point  Po'  is 
obtained,  which  hes  on  one  side  of  the  maximum. 

With  imsymmetrical  curves,  the  method  of  interpolation 
thus  does  not  give  the  exact  extreme.  For  most  engineering 
purposes  this  is  rather  an  advantage.  The  purpose  of  deter- 
mining the  extreme  usually  is  to  select  the  most  favorable 
operating  conditions.  Since,  however,  in  practice  the  operating 
conditions  never  remain  perfectly  constant,  but  vary  to  some 
extent,  the  most  favorable  operating  condition  in  Fig.  54  is 
not  that  where  the  average  value  gives  the  maximum  efficiency 


96— 

^ 

£^ 

_^ 

^ 

"^ 

^, 

90^- 

<u 
o 

88-53- 

Q. 

8a^ 

^ 

^ 

^ 

\ 

/ 

X 

S 

\ 

^ 

\ 

84-1 

\ 

\ 

\ 

\ 

\ 

0 

6 

fl 

~ 

n 

No 

8 

zzle 
0 

Ope 
9 

ling 
1 

0 

1 

1 

1 

2 

1.3 

Fig.  54.     Steam  Turbine  Nozzle  Efficiency;  Determination  of  Maximum. 

(point  Po),  but  the  most  favorable  operating  condition  is  that, 
where  the  average  efficiency  during  the  range  of  pressure,  occurr- 
ing in  operation,  is  a  maximum. 

If  the  steam  pressure,  and  thereby  the  required  expansion 
ratio,  that  is,  the  theoretically  correct  size  of  nozzle  mouth, 
should  vary  during  operation  by  25  per  cent  from  the  average, 
when  choosing  the  maximum  efficiencj^  point  Po  as  average, 
the  efficiency  during  operation  varies  on  the  part  of  the  curve 
between  Pi  (91.4  per  cent)  and  P2  (85.2  per  cent),  thus  averaging 
lower  than  by  choosing  the -point  Po'(6.25  per  cent  below  Po) 
as  average.  In  the  latter  case,  the  efficiency  varies  on  the 
part  of  the  curve  from  the  Pi'(90.1  per  cent)  to  P2'(90.1  per 
cent).     (Fig.  55.) 


152 


ENGINEERING  MATHEMATICS. 


Thus  in  ai)paratus  design,  when  determining  extrema  of 
a  function  y=J\x),  to  select  them  as  operating  condition, 
consideration  must  be  given  to  the  shape  of  the  curve,  and 
where  the  curve  is  unsymmetrical,  the  most  efficient  operating 
point  may  not  he  at  the  extreme,  but  on  that  side  of  it  at  which 
the  curve  falls  off  slower,  the  more  so  the  greater  the  range  of 
variation  is,  which  may  occur  during  operation.  This  is  not 
always  realized. 

loo.  If  the  function  y=f(x)  is  plotted  as  a  curve,  Fig. 
50,  at  the  extremes  of  the  function,  the  points  Pi,  P2,  Ps,  P4 
of  curve  Fig.  50,  the  tangent  on  the  curve  is  horizontal,  since 


90— 

Po' 

. < 

^ 

94 

P,    -. 

'■^ 

N. 

92 

90-| 

88-5- 
a. 

H 

82^ 

^ 

7^ 

V 

^ 

/ 

^ 

\ 

^ 

\ 

> 

b 

\ 

• 

\ 

\ 

0 

.fi 

0 

.7 

0 

No 

8 

:zle 
0 

Dper 
.9 

ing 
I 

,0 

1 

.1 

1 

.2 

J 

Fig.  55.     Steam  Turbine  Nozzle  Efficiency;  Determination  of  Maximum. 


at  the  extreme  the  function  changes  from  rising  to  decreasing 
(maximum,  Pi  and  P2),  or  from  decreasing  to  increasing  (min- 
imum, P3  and  P4),  and  th(u-efore  for  a  moment  passes  through 
the  horizontal  direction. 

In  general,  the  tangent  of  a  curve,  as  that  in  Fig.  50,  is  the 
line  which  connects  two  points  P'  and  P"  of  the  curve,  which 
are  infinitely  close  together,  and,  as  seen  in  Fig.  50,  the  angle 
6,  which  this  tangent  P'P"  makes  with  the  horizontal  or  X-axis, 
thus  is  given  by : 


tan  6- 


P"Q 
P'Q 


dy 
dx 


MAXIMA   AND  MINIMA.  153 

At  the  extreme,  the  tangent  on  the  curve  is  horizontal, 
that  is,  2^6  =  0,  and,  therefore,  it  follows  that  at  an  extreme 
of  the  function, 

y-m, (1) 

'i- (2) 

The  reverse,  however,  is  not  necessarily  the  case;    that  is, 

dy 
if  at  a  point  x,  y  :  -^  =  0,  this  point  may  not  be  an  extreme; 

that  is,  a  maximum  or  minimum,  but  may  be  a  horizontal 
inflection  point,  as  points  P5  and  Pq  are  in  Fig.  50. 

With  increasing  x,  when  passing  a  maximum  (Pi  and  Po, 
Fig.  50),  y  rises,  then  stops  rising,  and  then  decreases  again. 
When  passing  a  minimum  (P3  and  P4)  y  decreases,  then  stops 
decreasing,  and  then  increases  again.  When  passing  a  horizontal 
inflection  point,  y  rises,  then  stops  rising,  and  then  starts  rising 
again,  at  P5,  or  y  decreases,  then  stops  decreasing,  but  then 
starts  decreasing  again  (at  Pq). 

The  points  of  the  function  y=f{x),  determined  by  the  con- 

dy 
dition,  -f=0,  thus  require  further  investigation,  whether  they 

represent  a  maximum,  or  a  minimum,  or  merely  a  horizontal 
inflection  point. 

This  can  be  done  mathematically:  for  increasing  x,  when 
passing  a  maximum,  tan  d  changes  from  positive  to  negative; 

that  is,  decreases,   or  in    other  words,  -7-  (tan  6)<0.     Since 

dy  .  .  d^y 

tan  ^  =  "1^,  it  thus  follows  that  at  a  maximum  -p^  <  0.    Inversely, 

at  a  mininumi  tan  6  changes  from  negative  to  positive,  hence 

d  d^ll 

increases,  that  is,  -7-  (tan  ^)  >  0;  or,  -7-^2  ^  ^'-     When   passing 

a  horizontal   inflection   point  tan  6  first    decreases  to  zero  at 

the  inflection  point,  and  then  increases  again;    or,  invensely, 

tan  6  first  increases,  and  then  decreases  again,  that  is,  tan  d  = 

dy  .  .  . 

J-  has  a  maximum  or  a  minimum  at  the  inflection  point,  and 

d  d^y 

therefore,  -r-  (tan  '9)  =  t^  =  0  at  the  inflection  point. 


154  ENGINEERING  MATHEMATICS. 

In  engineering  problems    the   investigation,   whether  the 

dy 
solution   of   the   condition   of   extremes,   -r=0,   represents   a 

minimum,  or  a  maximum,  or  an  inflection  point,  is  rarely 
required,  but  it  is  almost  always  obvious  from  the  nature  of 
the  problem  whether  a  maximum  of  a  mmimum  occurs,  or 
neither. 

For  instance,  if  the  problem  is  to  determine  the  speed  at 
which  the  efficiency  of  a  motor  is  a  maximum,  the  solution: 
speed  =  0,  obviously  is  not  a  maxmium  but  a  mimimum,  as  at 
zero  speed  the  efficiency  is  zero.  If  the  problem  is,  to  find 
the  current  at  which  the  output  of  an  alternator  is  a  maximum, 
the  solution  i  =  0  obviously  is  a  minimum,  and  of  the  other 
two  solutions,  ii  and  io,  the  larger  value,  i2,  again  gives  a 
minimum,  zero  output  at  short-circuit  current,  while  the  inter- 
mediate value  ii  gives  the  maximum. 

loi.  The  extremes  of  a  function,  therefore,  are  determined 
by  equatmg  its  differential  quotient  to  zero,  as  is  illustrated 
by  the  following  examples : 

Example  4.  In  an  impulse  turbine,  the  speed  of  the  jet 
(steam  jet  or  water  jet)  is  *Si.  At  what  peripheral  speed  S2  is 
the  output  a  maximum. 

The  impulse  force  is  proportional  to  the  relative  speed  of 
the  jet  and  the  rotating  impulse  wheel;  that  is,  to  {S1-S2). 
The  power  is  impulse  force  times  speed  ^2;   hence, 

P  =  kS2(S^-S2), (3) 

dP 
and  is  an  extreme  for  the  value  of  ^2,  given  by  -r>-  =0;   hence, 

do  2 

Si-252  =  0    and     So  =  ~; (4) 


that  is,  when  the  peripheral  speed  of  the  impulse  wheel  equals 
half  the  jet  velocity. 

Example  5.  In  a  transformer  of  constant  impressed 
e.m.f.  eo  =  2300  volts;  the  constant  loss,  that  is,  loss  which 
is  independent  of  the  output  (iron  loss),  is  Pj  =  500  watts.  The 
internal  reL.istance  (primary  and  secondary  combined)  is  r  =  20 


MAXIMA  AND  MINIMA.  155 

ohms.     At  what  current  i  is  the  efficiency  of  the  transformer 
a  maximum;   that  is,  the  percentage  loss,  X,  a  minimum? 

ThelossisP  =  P,-+n2  =  500+20i2 (5) 

The  power  input  is  Pi  =ez  =  23001 ;  .     ...  (6) 
hence,  the  percentage  loss  is, 

;  =  Z  =  Zi±r?  (7) 


and  this  is  an  extreme  for  the  value  of  current  i,  given  by 
hence, 


^^  =  0- 
di       ' 


(Pi+ri^)e-ei(2ri) 


e^v 


or, 


Pt  — ri-=0     and     i  =  ^/— =  5  amperes,   ...     (8) 


and  the  output  is  Po  =  ex  =  11,500  watts.  The  loss  is,  P  =  P^  + 
n2  =  2P^  =  1000  watts;  that  is,  the  i^r  loss  or  variable  loss,  is 
equal  to  the  constant  loss  P^.     The  percentage  loss  is, 

P      vTr 

/I  =  p-  = =  0.087  =  8.7  per  cent, 

I  i  C 

and  the  maximum  efficiency  thus  is, 

l->i  =  0.913  =  91 .3  per  cent. 

102.  Usually,  when  the  problem  is  given,  to  determine 
those  values  of  x  for  which  y  is  an  extreme,  y  cannot  be  expressed 
directly  as  function  of  x,  y=f{x),  as  was  done  in  examples 
(4)  and  (5),  but  y  is  expressed  as  function  of  some  other  quan- 
ties,  y=fiu,  v  .  .),  and  then  equations  between  u,  v  . .  and  x 
are  found  from  the  conditions  of  the  problem,  by  w'hich  expres- 
sions of  X  are  substituted  for  k,  v  .  .,  as  shown  in  the  following 
example : 

Example  6.  There  is  a  constant  current  io  through  a  cir- 
cuit  containing  a   resistor  of   resistance   Tq.     This   resistor  ro 


156  ENGINEERING  MATHEMATICS. 

is  shunted  by  a  resistor  of  resistance  r.  What  must  bo  the 
resistance  of  this  shunting  resistor  r,  to  make  the  power  con- 
sumed in  r,  a  maximum?     (Fig.  56.) 

Let  i  be  the  current  in  the  shunting  resistor  r.  The  power 
consumed  in  r  then  is, 

P  =  ri^ (9) 

The  current  in  the  resistor  tq  is  io  —  h  and  therefore  the 
voHage  consumed  by  tq  is  ro(io—i),  and  the  voUage  consumed 
by  /'  is  ri,  and  as  these  two  vohages  must  be  equal,  since  both 


Fig.  56.     Shunted  Resistor. 

resistors  are  in  shunt  with  each  other,  thus  receive  the  same 
voltage^ 

n  =  ro(^o-^■), 


^b^ ) 


and,  herefrom,  it  follows  that. 


«=7Tk'" (10) 


Substituting  this  in  equation  (9)  gives, 

p=^:^j^  (11) 

dP 
and  this  power  is  an  extreme  for  -—  =  0;  hence: 

(r  +  ro)^roHn^  -  rroH(r2(r  +  rg) 

(r  +  ro)4  ' 

hence, 

r  =  ro; (12) 

that  is,  the  power  consumed  in  r  is  a  maxinmm,  if  the  resistor 
r  of  the  shunt  equals  the  resistance  ro. 


MAXIMA   AND  MINIMA.  157 

The.  current  in  r  then  is,  by  equation  (10), 


and  the  power  is, 


^  =  ^, 


4    • 


103.  If,  after  the  function  y=f{x)  (the  equation  (11)  in 
example   (6))  has  been  derived,  the  differentiation   -r  =  0    is 

immediately  carried  out,  the  calculation  is  very  frequently 
much  more  complicated  than  necessary.  It  is,  therefore, 
advisable  not  to  differentiate  immediately,  but  first  to  simplify 
the  function  y=f(x). 

If  y  is  an  extreme,  any  expression  differing  thereform  by 
a  constant  term,  or  a  constant  factor,  etc.,  also  is  an  extreme. 
So  also  is  the  reciprocal  of  y,  or  its  square,  or  square  root,  etc. 

Thus,  before  differentiation,  constant  terms  and  constant 
factors  can  be  dropped,  fractions  inverted,  the  expression 
raised  to  any  power  or  any  root  thereof  taken,  etc. 

For  instance,  in  the  preceding  example,  in  equation  (11), 

rr  oHo^ 


(r  +  ro)-" 

the  value  of  r  is  to  be  found,  which  makes  P  a  maximum. 
If  P  is  an  extreme, 

r 

'^'^  {r  +  ror 

which  differs  rrom  P  by  the  omission  of  the  constant  factor 
tqHo^,  also  is  an  extreme. 
The  reverse  of  i/i, 

(r  +  ro)2 


2/2 


r 


is  also  an  extreme.     (1/2  is  a  minimum,  where  yi  is  a  maximum, 
and  inversely.) 

Therefore,  the  equation  (11)  can  be  simplified  to  the  form  : 

(r+ro)2  ro2 

2/2= —  =  r+2ro+— , 


158  ENGINEERING  MATHEMATICS. 

and,  leaving  out  the  constant  term  2ro,  gives  the  final  form, 


?/3  =  r+~. (13) 


This  differentiated  gives, 
hence, 


dr  f^        ' 


104.  Example  7.  From  a  source  of  constant  alternating 
e.m.f.  e,  power  is  transmitted  over  a  line  of  resistance  tq  and 
reactance  xo  into  a  non-inductive  load.  What  must  be  the 
resistance  r  of  this  load  to  give  maximum  power? 

If  I  =  current  transmitted  over  the  line,  the  power  delivered 
at  the  load  of  resistance  r  is 

P  =  n2 (14) 

The  total  resistance  of  the  circuit  is  r-\-ro;  the  reactance 
is  Xq]   hence  the  current  is 

1  =  —^=^=, (15) 

V  (r  +  ro)2+.ro^ 

and,  by  substituting  in  equation  (14),  the  power  is 

p^ -, (10) 

if  P  is  an  extreme,  by  omitting  e^  and  inverting, 

(r  +  ro)2+Xo2 

yy= 

r      ' 
is  also  an  extreme,  and  likewise, 


*  T 


is  an  extreme. 


MAXIMA  AND  MINIMA.  159 

Differentiating,  gives: 

dr  f^  ' 


r  =  \W+x^.        (17) 

"WTierefrom  follows,  by  substituting  in  equation  (16), 


irp  +  Vro^+Xo^)^  +  Xo^ 


2{ro  +  Vrp^+Xp^) 


(18) 


Very  often  the  function  y=f{x)  can  by  such  algebraic 
operations,  which  do  not  change  an  extreme,  be  simplified  to 
such  an  extent  that  differentiation  becomes  entirely  unnecessary, 
but  the  extreme  is  immediately  seen;  the  following  example 
will  serve  to  illustrate : 

Example  8.  In  the  same  transmission  circuit  as  in  example 
(7),  for  what  value  of  r  is  the  current  i  a  maximum? 

The  current  i  is  given,  by  equation  (15), 


i  =  - 


Dropping  e  and  reversing,  gives. 


1/1  =  V(r+ro)2+Xo2; 
Squaring,  gives, 

2/2  =  (r  +  ro)2  4-xo2; 

dropping  the  constant  term  xp^  gives 

2/3  =  (r+ro)2;  (19) 

taking  the  square  root  gives 

^4  =  r+ro; 


160  ENGINEERING  MATHEMATICS. 

dropping  the  constant  term  ro  gives 

y5=n (20) 

that  is,  the  current  i  is  an  extreme,  when  11/5  =  r  is  an  extreme, 
and  this  is  the  case  for  r  =  0  and  r  =  oo  :   r  =  0  gives, 

i=     .-^^ (21) 

as  the  maximum  vakie  of  the  ciu-rent,  and  r=oo  gives 

1  =  0, 

as  the  minimum  vahie  of  the  current. 

With  some  practice,  from  the  original  equation  (1),  imme- 
diately, or  in  ver}^  few  steps,  the  simplified  final  equation  can 
be  derived. 

105.  In  the  calculation  of  maxima  and  minima  of  engineer- 
ing quantities  x,  y,  by  difTerentiation  of  the  function  y=f(x), 
it  must  be  kept  in  mind  that  this  method  gives  the  values  of 
z,  for  which  the  quantity  y  of  the  mathematical  equation  y  =f{x) 
becomes  an  extreme,  but  whether  this  extreme  has  a  physical 
meaning  in  engineering  or  not  requires  further  investigation; 
that  is,  the  range  of  numerical  values  of  x  and  y  is  unUmited 
in  the  mathematical  equation,  but  may  be  limited  in  its  engineer- 
ing application.  For  instance,  if  x  is  a  resistance,  and  the 
differentiation  of  y=f(x)  leads  to  negative  values  of  x,  these 
have  no  engineering  meaning;  or,  if  the  differentiation  leads 
to  values  of  x,  which,  substituted  in  y=f{x),  gives  imaginary,  or 
negative  values  of  y,  the  result  also  may  have  no  engineering 
appUcation.  In  still  other  cases,  the  mathematical  result 
may  give  values,  which  are  so  far  beyond  the  range  of  indus- 
trially practicable  numerical  values  as  to  be  inapplicable. 
For  instance : 

Example  9.  In  example  (8),  to  determine  the  resistance 
r,  which  gives  maximum  current  transmitted  over  a  trans- 
mission line,  the  equation  (15), 


V(r+ro)2+xo2' 


MAXIMA   AND  MINIMA.  161 

immediately  differentiated,  gives  as  condition  of  the  extremes: 
rfi_ 2(r+ro) 

hence,  cither  r  +  ro  =  0; (22) 

or,  (r+ro)2+xo2  =  oc (23) 

the  latter  equation  gives  r  =  oo;   hence  i  =  0,  the  minimum  value 
of  current. 

The  former  equation  gives 

r=-ro, (24) 

as  tne  value  of  the  resistance,  which  gives  maximum  current, 
and  the  current  would  then  be,  by  substituting  (24)  into  (15), 

^=7 (25) 

Xq 

The  solution  (24),  however,  has  no  engineering  meaning, 
as  the  resistance  r  cannot  be  negative. 

Hence,  mathemetically,  there  exists  no  maximum  value 
of  I  in  the  range  of  r  which  can  occur  in  engineering,  that  is, 
within  the  range,  0<  r<  oo. 

In  such  a  case,  where  the  extreme  falls  outside  of  the  range 
of  numerical  values,  to  which  the  engineering  quantity  is 
limited,  it  follows  that  within  the  engineering  range  the  quan- 
tity continuously  increases  toward  one  limit  and  continuously 
decreases  tow^ard  the  other  limit,  and  that  therefore  the  two 
Umits  of  the  engineering  range  of  the  quantity  give  extremes. 
Thus  r  =  0  gives  the  maximum,  r  =  oothe  minimum  of  current. 

io6.  Example  lo.  An  alternating-current  generator,  of 
generated  e.m.f.  e  =  2500  volts,  internal  resistance  ro  =  0.25 
ohms,  and  synchronous  reactance  0*0  =  10  ohms,  is  loaded  by 
a  circuit  comprising  a  resistor  of  constant  resistance  r  =  20 
ohms,  and  a  reactor  of  reactance  x  in  series  with  the  resistor 
r.     What  value  of  reactance  x  gives  maximum  output? 

If  I  =  current  of  the  alternator,  its  power  output  is 

P  =  n2  =  20i2; (26) 


162  ENGINEERING  MATHEMATICS. 

the  total  resistance  is  r  +  ro  =  20.25  ohms;   the  total  reactance 
is  x+xo  =  10+j-  ohms,  and  therefore  the  current  is 

....     (27) 


V(r+ro)2  +  (x+.xo)2' 
and  the  power  output,  by  substituting  (27)  in  (26),  is 

p_  re^  _       20X2500^ 

(r  +  ru^)+(x  +  Xo)2     20.252  + (10 +a:)2-  •     •     V-^' 

Simplified,  this  gives 

2/i  =  (r  +  ro)2  +  (x+Xo)2; (29) 

i/2---(x+Xo)2; 
hence, 

^  =  2(..+.„)=0; 

and 

X  =  —  .To  =  —  ]  0  ohms : (30) 

that  is,  a  negative,  or  condensive  reactance  of  10  ohms.     The 
power  output  would  then  be,  by  substituting  (30)  into  (28), 

re2         20+25002  ^^^  , 

watts  =  305  kw.    .     .     (31) 


(r  +  ro)2         20.252 

If,  however,  a  condensive  reactance  is  excluded,  that  is, 
it  is  assumed  that  x  >0,  no  mathematical  extreme  exists  in  the 
range  of  the  variable  x,  which  is  permissible,  and  the  extreme 
is  at  the  end  of  the  range,  x  =  0,  and  gives 

P=<    /^i^    2-245  kw (32) 

(r  +  ro)2+xo2 

107.  Example  11.  In  a  500-kw.  alternator,  at  voltage 
e  =  2500,  the  friction  and  windage  loss  is  -P„,  =  6  kw.,  the  iron 
loss  Pi  =  24  kw.,  the  field  excitation  loss  is  Py  =  6  kw.,  and  the 
armature  resistance  r  =  0.1  ohm.  At  what  load  is  the  eflficiency 
a  maximum? 


MAXIMA  AND  MINIMA.  163 

The  sum  of  the  losses  is: 

P  =  p^  +  p. +P^  +  n2^  36,000+0.172.     .     .     .     (33) 

The  output  is 

Po^ei=^2500i; (34) 

hence,  the  efficiency  is 

Po  ei  2500?: 


'^     Po+P     ei+P^+Pi  +  P/-{-rP    36000 +25001 +0.l2;2' 
or,  simphfied, 


(35) 


hence, 
and, 


P^  +  Pi  +  Pf^    . 
r/i= -. +  n; 


^=       Pw  +  Pj  +  Pf 
di     ^  i^ 


.       \P^  +  Pi  +  Pf_    /36000 

1  =  *  / ~  ^    ■-.  -.  ■  =  oOO  amperes, 


(36) 


and  the  output,  at  which  the  maximum  efficiency  occurs,  by 
substituting  (36)  into  (34),  is 

p  =  et  =  1500kw., 

that  is,  at  three  times  full  load. 

Therefore,  this  value  is  of  no  engineering  importance,  but 
means  that  at  full  load  and  at  all  practical  overloads  the 
maximum  efficiency  is  not  yet  reached,  but  the  efficiency  is 
still  rising. 

io8.  Frequently  in  engineering  calculations  extremes  of 
engineering  quantities  are  to  be  determined,  which  are  func- 
tions or  two  or  more  independent  variables.  For  instance, 
the  maximum  power  is  required  which  can  be  delivered  over  a 
transmission  line  into  a  circuit,  in  which  the  resistance  as  well 
as  the  reactance  can  be  varied  independently.  In  other 
words,  if 

y=Ku,v) (37) 


164  ENGINEERING  MATHEMATICS. 

is  a  function  of  two  independent  variables  u  and  v,  such  a 
pair  of  values  of  u  and  of  v  is  to  be  found,  which  makes  y  a 
maximum,  or  minimum. 

Choosing  any  value  uq,  of  the  independent  variable  u, 
then  a  value  of  v  can  be  found,  w^hich  gives  the  maximum  (or 
mininmm)  value  of  y,  which  can  be  reached  for  u  =  uq.  This 
is  done  by  differentiating  y=f{uo,v),  over  v,  thus: 

•^^^  =  0 (38) 

From  this  equation  (38),  a  value, 

r=/i(uo), (39) 

is  derived,  which  gives  the  maximum  value  of  y,  for  the  given 
value  of  Wo,  and  by  substituting  (39)  into  (38), 

y=f2iu.o), (40) 

is  obtained  as  the  equation,  which  relates  the  different  extremes 
of  y,  that  correspond  to   the  different  values  of  Wo,  with    uq. 
Herefrom,  then,  that  \'alue  of  uo  is  found  which  gives  the 
maximum  of  the  maxima,  by  differentiation: 

^^0 (41) 

Geometrically,  y=f(u,v)  may  be  represented  by  a  surface 
in  space,  with  the  coordinates  y,  u,  v.  y  =f{uo,v),  then,  represents 
the  curve  of  intersection  of  this  surface  with  the  plane  uo  = 
constant,  and  the  differentation  gives  the  maximum  point 
of  this  intersection  curve.  y=f 2(110)  then  gives  the  curve 
in  space,  which  connects  all  the  maxima  of  the  various  inter- 
sections with  the  Wo  planes,  and  the  second  differentiation 
gives  the  maximum  of  this  maximum  curve  y^f-ziuo),  or  the 
maximum  of  the  maxima  (or  more  correctly,  the  extreme  of 
the  extremes). 

Inversely,  it  is  possible  first  to  differentiate  over  u,  thus, 


MAXIMA  AND  MINIMA.  165 

and  thereby  get 

w=/3(i'o). (43) 

as  the  vahic  of  m,  which  makes  y  a  maximum  for  the  given 
vakie  of  v  =  Vo,  and  substituting  (43)  into  (42), 

y=f 4(vo), (44) 

is  obtained  as  the  equation  of  the  maxima,  which  differentiated 
over  vo,  thus, 

*^=0, (45) 

gives  the  maximum  of  the  maxima. 

Geometrically,  this  represents  the  consideration  of  the 
intersection  curves  of  the  surface  with  the  planes  t;  =  constant. 

However,  equations  (38)  and  (41)  (respectively  (42)  and 
(45))  give  an  extremum  only,  if  both  equations  represent 
maxima,  or  both  minima.  If  one  of  the  equations  represents 
a  maximum,  the  other  a  minimum,  the  point  is  not  an  extre- 
mum, but  a  saddle  point,  so  called  from  the  shape  of  the  sur- 
face y=f(u,  v)  near  this  point. 

The  working  of  this  will  be  plain  from  the  following  example : 

lOQ.  Example  12.  The  alternating  voltage  e  =  30,000  is 
impressed  upon  a  transmission  line  of  resistance  ro  =  20  ohms 
and  reactance  .ro  =  50  ohms. 

What  should  be  the  resistance  r  and  the  reactance  x  of  the 
receiving  circuit  to  deliver  maximimi  power? 

Let  t  =  current  delivered  into  the  receiving  circuit.  The 
total  resistance  is  (r  +  ro);  the  total  reactance  is  (x+Xo);  hence, 
the  current  is 

i=    ,  ^     (46) 

The  power  output  is         P  =  ri^;        (47) 

hence,  substituting  (46),  gives 

^^ir+ro)^  +  (x+XQ)2 ^^^^ 

(a)  For  any  given  value  of  r,  the  reactance  x,  which  gives 

•     ,     .      ,  ,      dP    ^ 
maximum  power,  is  derived  by  -r-  =  0. 

ax 


166  ENGINEERING  MATHEMATICS. 

P  simplified,  gives  yi  =  {x^-XqY]  hence, 

-^=2(x+Jo)=0    and     x=-Xo    .     .     .     (49) 
ax 

that  is,  for  any  chosen  resistance  r,  the  power  is  a  maximum, 
if  the  reactance  of  the  receiving  circuit  is  chosen  equal  to  that 
of  the  Une,  but  of  opposite  sign,  that  is,  as  condensive  reactance. 
Substituting  (49)  into  (48)  gives  the  maximum  power 
available  for  a  chosen  value  of  r,  as : 


or,  simplified, 
hence, 


(r  +  ro)2  ro^ 

y2  =  —^ and      t/3  =  r+Y 


—  =l—^      and         r  =  ro,     ....     (51) 


and  by  substituting  (51)  into  (50),  the  maximum  power  is, 

6-2 


(52) 


(6)  For  any  given  value  of  x,  the  resistance  r,  which  gives 

maximum  power,  is  given  by  -j—  =  0. 

P  simpUfied  gives, 

{r+ro)'^  +  {x+Xo)^  ,  ro''  +  {x+Xo)^ 

2/1  = ;       2/2-=r  + ; 

r  r 

dyo  _ .     ro^  +  {x  +  X())^_ 
dr      ^  r-' 


r=Vro^  +  (x  +  xoY,        .    ••     .     .     .     (53) 

which  is  the  value  of  r,  that  for  any  given  value  of  x,  gives 
maximum  power,  and  this  maximum  power  by  substituting 
(53)  into  (48)  is, 

o  vVn2  +  (j  +  xo^V 


[ro  +  Vro^  +  (X  +  xoyf+  (x  +  xo)^ 

....     (54) 


2iro  +  \/ro2  +  (x  +  Xo)2{ 


MAXIMA  AND  MINIMA.  167 

which  is  the  maximum  power  that  can  be  transmitted  into  a 
receiving  circuit  of  reactance  x. 

The  value  of  x,  which  makes  this  maximum  power  Pq  the 

highest  maximum,  is  given  by  —, —  =0. 
Pq  simpUfied  gives 


?/3  =  ro  +  v/ro2  +  (x+xo)2; 


2/4=  Vro^  +  (x+xo)^; 

i/6  =  (-r+Xo)2; 

?/7  =  (x+xo); 

and  this  value  is  a  maximum  for  (j+.to)=0;  that  is,  for 

x=-xo (55) 

Note.  If  x  cannot  be  negative,  that  is,  if  only  inductive 
reactance  is  considered,  x  =  0  gives  the  maximum  power,  and 
the  latter  then  is 

^  m&x  —  ~       1^/9,  ^tT'       ....     (56) 
2jro  +  vro2+Xo^l 

the  same  value  as  found  in  problem  (7),  equation  (18). 

Substituting  (55)  and  (54)  gives  again  equation  (52),  thus, 

P      =^ 

1 10.  Here  again,  it  requires  consideration,  whether  the 
solution  is  practicable  within  the  Umitation  of  engineering 
constants. 

With  the  numerical  constants  chosen,  it  would  be 

e2      300002 
^max  =  4^^  =  —gQ-  =11,250  kw.; 

e 
"■  =  K-  =  750  amperes. 


168  ENGINEERING  MATHEMATICS. 

and  the  voltage  at  the  receiving  end  of  the  Une  would  be 


e' = I  \/ r2  +  x2  =  7oO\/2(F  +  5U2  =  40,400  volts ; 

that  is,  the  voltage  at  the  receiving  end  would  be  far  higher 
than  at  the  generator  end,  the  current  excessive,  and  the  efficiency 
of  transmission  only  50  per  cent.  This  extreme  case  thus  is 
hardly  practicable,  and  the  conclusion  would  be  that  by  the 
use  of  negative  reactance  in  the  receiving  circuit,  an  amount 
of  power  could  be  delivered,  at  a  sacrifice  of  efficiency,  far 
greater  than  economical  transmission  would  permit. 

In  the  case,  where  capacity  was  excluded  from  the  receiv- 
ing circuit,  the  maximum  power  was  given  by  equation  (56)  as 

III.  Extremes  of  engineering    quantities  x,  y,  are  usually 
determined  by  differentiating  the  function. 


and  from  the  equation, 


y=m, (57) 

I'^O (58) 


deriving  the  values  of  x,  which  make  y  an  extreme. 

Occasionally,  however,  the  eciuation  (58)  cannot  be  solved 
for  X,  but  is  either  of  higher  order  in  x,  or  a  transcendental 
equation.  In  this  case,  equation  (58)  may  be  solved  by  approx- 
imation, or  preferably,  the  function, 

e  =  ^ (59) 

dx, 

is  plotted  as  a  curve,  the  values  of  .r  taken,  at  which  z  =  0, 

that  is,  at  which  the  curve  intersects  the  X-axis.     For  instance : 

Example    13.    The    e.m.f.  wave   of  a  three-])hase  alternator, 

as  determined  by  oscillograph,  is  represented  by  the  equation, 

e  =  36000{sin  (9-0.12  sin  (3^-2.3°)-23  sin  (5^-1.5°) + 

0.13  sin  (7^-0.2°)! (CO) 


MAXIMA   AND  MINIMA. 


169 


This  alternator,  connected  to  a  long-distance  transmission  line, 
gives  the  charging  current  to  the  line  of 

1  =  13.12  cos  (^-0.3°) -5.04  cos  (3^- 3.3°) -18.76  cos  (5^- 3.6°) 

+  19.59  cos  (7^-9.9°)     ....     (61) 

(see  Chapter  III,  paragraph  95). 

What  are  the  extreme  values  of  this  current,  and  at  what 
phase  angles  6  do  they  occur? 

The  phase  angle  6,  at  which  the  current  i  reaches  an  extreme 
value,  is  given  by  the  equation 


^  =  0 


(62) 


Fig.  57. 


Substituting  (61)  into  (62)  gives, 

di 
2=^=  - 13.12  sin  (^-0.3°)  +15.12  sin  (3i9-3.3°)  +93.8  sin 

(5^-3.6°)-137.1  sin  (7^-9.9°)=0.     .     .     .     (63) 

This  equation  cannot  be  solved  for  6.  Therefore  z  is 
plotted  as  function  of  6  by  the  curve.  Fig.  57,  and  from  this 
curve  the  values  of  6  taken  at  which  the  curve  intersects  the 
zero  line.     They  are: 

^  =  1°;  20°;  47°  78°;  104°;  135°;  162°. 


170  ENGINEERING  MATHEMATICS. 

For  these  angles  d,  the  corresponding  values  of  i  are  calculated 
by  equation  (61),  and  are: 

10= -1-9;   _i;    +39;   -30;    +30;   -42;    +4amperes. 

The  current  thus  has  during  each  period  14  extrema,  of 
which  the  highest  is  42  amperes. 

112.  In   those   cases,   where   the   mathematical   expression 

of  the  function  y=f{x)  is  not  know^n,  and  the  extreme  values 

therefore  have  to  be  determined  graphically,  frequently  a  greater 

accuracy  can  be  reached  by  plotting  as  a  curve  the  differential 

of  y=f{x)  and  picking  out  the  zero  values  instead  of  plotting 

y=f{x),  and  picking  out  the  highest  and  the  lowest  points. 

If  the  mathematical  expression  of  y=f(x)  is  not  known,  obvi- 

dv 
ously  the  equation  of  the  differential  curve  z=-j-  (64)  is  usually 

not  known  either.  Approximately,  however,  it  can  fre- 
quently be  plotted  from  the  numerical  values  of  y=f{x),  as 
follows :  ' 

If      x\,  X2,  Xa  .  .  .  are  successive  numerical  values  of  x, 

and         y\,  1/2,  2/3  ••  .  the  corresponding  numerical  values  of  y, 

dv 
approximate    points   of  the  differential  curve  z=-r  are  ^ven 

by  the  corresponding  values: 

X2+xx       xz-\-X2      a-4-f-a:3 


as  abscissas : 
as  ordinates: 


2      '  2      '  2 

X2  —  X1  X3~2*2  '       X^—Xs 


113.  Example  14,  In  the  problem  (1),  the  maximum  permea- 
bility point  of  a  sample  of  iron,  of  which  the  B,  H  curve  is  given 
as  Fig.  51,  was  determined  by  taking  from  Fig.  51  corresponding 

values  of  B  and  H,  and  plotting  t^  =  j},  against  B  in    Fig.  52. 

A  considerable  inaccuracy  exists  in  this  method,  in  locating 
the  value  of  B,  at  which  //  is  a  maximum,  due  to  the  flatness 
of  the  curve.  Fig.  52. 


MAXIMA  AND  MINIMA. 


171 


The  successive  pairs  of  corresponding  values  of  B  and  H, 
as  taken  from  Fig.  51  are  given  in  columns  1  and  2  of  Table  I. 

Table   I. 


B 

Kilolinea, 

H 

B 

J/z 

B 

0 

0 

370 

1 

1.76 

570 

+  200 

0.5 

2 

2.74 

730 

160 

1.5 

3 

3.47 

865 

135 

2.5 

4 

4.06 

985 

120 

3.5 

5 

4.59 

1090 

105 

4.5 

6 

5.10 

1175 

85 

5.5 

7 

5.63 

1245 

70 

6.5 

8 

6.17 

1295 

50 

7.5 

9 

6.77 

1330 

35 

8.5 

10 

7.47 

1340 

+  10 

9.5 

11 

8.33 

1320 

-20 

10.5 

12 

9.60 

1250 

70 

11.5 

13 

11.60 

1120 

130 

12.5 

14 

15.10 

930 

190 

13.5 

15 

20.7 

725 

205 

14.5 

In  the  third  column  of  Table  I  is  given  the  permeability, 

/x  =— ,  and  in  the  fourth  column  the  increase  of  permeability, 
H 

per  B  =  l,  ^/i;  the  last  column  then  gives  the  value  of  B,  to 
which  i/i  corresponds. 

In  Fig.  58,  values  J/z  are  plotted  as  ordinates,  with  B  as 
abscissas.     This  curve  passes  through  zero  at  B=9.95. 

The  maximum  permeability  thus  occurs  at  the  approximate 
magnetic  density  B=9.95  kilolines  per  sq.cm.,  and  not  at  B= 
10.2,  as  was  given  by  the  less  accurate  graphical  determination 
of  Fig.  52,  and  the  maximum  permeability  is  //()  =  1340. 

As  seen,  the  sharpness  of  the  intersection  of  the  differential 
curve  with  the  zero  line  permits  a  far  greater  accuracy  than 
feasible  by  the  method  used  in  instance  (1). 

114.  As  illustration  of  the  method  of  determining  extremes, 
some  further  examples  are  given  below: 


172 


ENGINEERING  MATHEMATICS. 


Example  15.  A  storage  battery  of  n  =  80  cells  is  to  be 
connected  so  as  to  give  maximum  power  in  a  constant  resist- 
ance r  =  0.1  ohm.  Each  battery  cell  has  the  e.m.f.  eo  =  2.1 
volts  and  the  internal  resistance  ro  =  0.02  ohm.  How  must 
the  cells  be  connected? 

Assuming  the  cells  are  connected  with  x  in  parallel,  hence 
n  . 
-  m  series.     The  internal  resistance  of  the   battery  then  is 

n 

-To 

X         nro    ,  ,   ,  ,  n 

=—Y  ohms,  and  the  total  resistance  of  the  circuit  is  — ,ro  +  r. 


Fig.  58.     First  Differential  Quotient  of  B,n  Curve 

71  71 

The  e.m.f.  acting  on  the  circuit  is  -eo,  since—  cells  of  e.m.f. 
*'  X     '  X 

eo  are  in  series.     Therefore,  the  current  delivered  by  the  battery 

is, 

n 
-^0 


ro  +  r 


X- 


and  the  power  which  this  current  ]iroduces  in  the  resistance 
r,  is. 


P  =  H^  = 


(72^0  +  r) 


MAXIMA  AND  MINIMA.  173 

This  is  an  extreme,  if 


is  an  extreme,  hence. 


and 


nro  , 
y= hrx 


ax         x^  ' 


x  = 


Vf°-; 


that   is,    ^=-yj —  =  4    cells   are    connected    in    multiple,    and 

Ti       iTir 

-  =  *  / —  =  20  cells  in  series. 

X     \ro 

115.  Example  16,  In  an  alternating-current  transformer  the 
loss  of  power  is  limited  to  900  watts  by  the  permissible  temper- 
ature rise.  The  internal  resistance  of  the  transformer  winding 
(primary,  plus  secondary  reduced  to  the  primary)  is  2  ohms, 
and  the  core  loss  at  2000  volts  impressed,  is  400  watts,  and 
varies  with  the  1 .6th  power  of  the  magnetic  density  and  there- 
fore of  the  voltage.  At  what  impressed  voltage  is  the  output 
of  the  transformer  a  maximum? 

If  e  is  the  impressed  e.m.f.  and  i  is  the  current  input,  the 
power  input  into  the  transformer  (approximately,  at  non- 
inductive  load)  is  P  =  eL 

If  the  output  is  a  maximum,  at  constant  loss,  the  input  P 
also  is  a  maximum.      The  loss  of  power  in    the  winding  is 

The  loss  of  power  in  the  iron  at  2000  volts  impressed  is 
400  watts,  and  at  impressed  voltage  e  it  therefore  is 


1-6 

V20OO 


I  — »      X400, 


and  the  total  loss  in  the  transformer,  therefore,  is 


174  ENGINEERING  MATHEMATICS. 

herefrom,  it  follows  that, 


»  =  ^450-200(^y", 
and,  substituting,  into  P=ei: 


Simplified,  this  gives. 


2/  =  2.25e2. 


,3-6 


20001-6' 

and,  differentiating, 

dy     ,  ,      3.6^2-6 

de  2000i<*    ^' 

and 

\2000/      ^■^•^^• 
Hence, 

^-^  =  1.15    and     e  =  2300  volts, 
which,  substituted,  gives 


P  =  2300\/450 -  200  X 1 .25  =  32.52  kw. 

ii6.  Example  17.  In  a  5-kw.  alternating-current  transformer, 
at  1000  volts  impressed,  the  core  loss  is  60  watts,  the  i^r  loss 
150  watts.  How  must  the  impressed  voltage  be  changed, 
to  give  maximum  efficiency,  (a)  At  full  load  of  5-kw;  (h)  at 
half  load? 

The  core  loss  may  be  assumed  as  varying  with  the  1.6th 

power  of  the  impressed  voltage.     If  e  is  the  impressed  voltage, 

.    5000  .     ,  .„,,,.      2500.     , 

I  = IS  the  current  at  full  load,  and  2i  = is  the  current  at 

e  '  e 

half  load,  then  at  1000  volts  impressed,  the  full-load  current  is 

5000       ,  ,      .  ,         .0     ^  '      .rr.  ,  .  . 

=  5  amperes,  and  smce  the  i^r  loss  is  150  watts,  this  gives 


MAXIMA  AND  MINIMA.  175 

the  internal  resistance  of  the  transformer  as  r  =  6  ohms,  and 
herefrom  the  i^r  loss  at  impressed  voltage  e  is  respectively, 

.,     150X106        ^       .^    37.5X106 
n^  = 5 and    ni^  = ^ watts. 

Since  the  core  loss  is  60  watts  at  1000  volts,  at  the  voltage  e 

it  is 

/    e   V'^ 
^i=60x(^YQ^j      watts. 

The  total  loss,  at  full  load,  thus  is 

D       D   ,    -9    ^n     /    e   \i-6     150X106 

and  at  half  load  it  is 

.5X106 


looo)    +-- 

Simplified,  this  gives 

!'=(K)i)o)"'+2.5X10exe- 


y'=\Tm) 


6 

+0.625X106X6-2; 


hence,  differentiated. 


1-^^1^-6-5X1066-3  =  0; 

^•^1^-1-''>^1^'>^'"'  =  ^' 
e3-6  =  3.125Xl06xl000i-6  =  3.125Xl0io8; 
e3-6  =  0.78125  X 106  X  lOOO^-e  =  0.78125  X  lO^o-^ ; 

hence,        e  =  1373  volts  for  maximum  efficiency  at  full  load. 

and  e  =  938  volts  for  maximum  efficiency  at  half  load. 

117.  Example  18.  (a)  Constant  voltage  e  =  1000  is  im- 
pressed  upon  a  condenser  of  capacity  C  =  10  mf.,  through  a 
reactor  of  inductance  L  =  100  mh.,  and  a  resistor  of  resist- 
ance r  =  40  ohms.  What  is  thp  maximum  value  of  the  charg- 
ing current 


176  ENGINEERING  MATHEMATICS. 

(b)  An  additional  resistor  of  resistance  r'  =  2\0  ohms  is 
then  inserted  in  series,  mailing  the  total  resistance  of  the  con- 
denser charging  circuit,  r  =  250  ohms.  What  is  the  maximum 
value  of  the  charging  current? 

The  equation  of  the  charging  current  of  a  condenser,  through 
a  circuit  of  low  resistance,  is  {"  Transient  Electric  Phenomena 
and  Oscillations,"  p.  61): 


.     2e     -^t   .      q 
i  =  -    £    2L'smrn-< 
q  2L 


where 


4L 

c 


r2, 


and  the  equation  of  the  charging  current  of  a  condenser,  through 
a  circuit  of  high  resistance,  is  ("  Transient  Electric  Phenomena 
and  Oscillations,"  p.  51), 

{  =  -{£      2L'_£      2L' 

where 


5  = 


Substituting  the  numerical  values  gives: 

(a)  ?:  =  10.2  £-20f»  sin  980  ^, 
(6)                      i  =  6.667 {  £-  500' _  s-20oot  j  _ 

Simplified  and  differentiated,  this  gives: 

(o)  2 =-^  =  4.9  cos  980^ -sin  980^  =  0; 

hence  tan  980/ =  4. 9 

980/ =  68.5°        =1.20 

1.20+nr 
^=      908       '''■ 

(b)  ;2  =  ^  =  4£-2ooo«_,-5oo/==0; 

at 


MAXIMA  AND  MINIMA.  177 


hence,  £+]5oo<  =  4^ 

log  4 
1500^  =  ,— =  1.38, 

log   £ 

f  =  0.00092  sec, 

and,  by  substituting  these  values  of  t  into  the  equations  of  the 
current,  gives  the  maximum  values: 

1.20  + UK 

(a)       i  =  10s      4.9     =7.83£-o-64«  =  7.83x0.53«  amperes; 

that  is,  an  infinite  number  of  maxima,  of  gradually  decreasing 
values:  +7.83;    -4.15;    +.2.20;    -1.17  etc. 

(6)  i  =  6.667 (£-0-46-  £-1.84)  _  3  16  amperes. 

ii8.  Example  19.  In  an  induction  generator,  the  fric- 
tion losses  are  P^=100  kw,;  the  iron  loss  is  200  k\v.  at  the  ter- 
minal voltage  of  e  =  4  kv.,  and  may  be  assumed  as  proportional 
to  the  1.6th  power  of  the  voltage;  the  loss  in  the  resistance 
of  the  conductors  is  100  kw.  at  i  ^  3000  amperes  output,  and  may 
be  assumed  as  proportional  to  the  square  of  the  current,  and 
the  losses  resulting  from  stray  fields  due  to  magnetic  saturation 
are  100  kw.  at  e  =  4  kv.,  and  may  in  the  range  considered  be 
assumed  as  approximately  proportional  to  the  3.2th  power 
of  the  voltage.  Under  what  conditions  of  operation,  regard- 
ing output,  voltage  and  current,  is  the  efficiency  a  maximum? 

The  losses  mav  be  summarized  as  follows: 


Friction  loss,  P/=100  kw.; 


Iron  loss,  P,+200., 

'  \4 


1-6 


Saturation  loss,  P^  =  100(  j  )     ; 

hence  the  total  loss  is  PL  =  r*f  +  Pi  +  Pc  +  Pg 


=^->h^4r<^j^{r 


178  ENGINEERING  MATHEMATICS. 

The  output  is  P  =  ei;  hence,  percentage  of  loss  is 

The  efficiency  is  a  maximum,  if  the  percentage  loss  -^  is  a 
minimum.      For  any  value  of  the  voltage  e,  this  is  the  case 

at  the  current  i,  given  by  -tt  =  0;  hence,  simpUfying  and  differ- 
entiating X, 


a      1+21. 


'  onnn2     ^ ' 


di  i^  30002 

i  =  3000^1+2(|)"V(|)'"'; 
then,  substituting  i  in  the  expression  of  X,  gives 

and  A  is  an  extreme,  if  the  simplified  expression, 
_!       2        _1_  ^2 

is  an  extreme,  at 

de        ^3    41-6^1.4 ^43.2*^   » 

hence,  2+5^e^^-6-^,e3-2=.0; 

/e\i-6      2 
hence,  (^)     =y9     and     e  =  5.50kv., 

and,  by  substitution  the  following  values  are  obtained :  X  =  0.0323 ; 
efficiency  96.77  per  cent;  current  t  =  8000  amperes;  output 
P  =  44,000  kw. 

119.  In  all  probability,  this  output  is  beyond  the  capacity 
of  the  generator,  as  limited  by  heating.  The  foremost  limita- 
tion probably  will  be  the  i^r  heating  of  the  conductors;  that  is, 


MAXIMA  AND  MINIMA.  179 

the  maximum  permissible   current  will   be  restricted  to,   for 
instance,  i  =  5000  amperes. 

For  any  given  value  of  current  i,  the  maximum  efficiency, 
that  is,  minimum  loss,  is  found  by  differentiating, 

ioo[i+o(iy%(-L.)%^l 

/  — 


ei 
over  e,  thus : 

de 
Simplified,  X  gives 

hence,  differentiated,  it  gives 

dy^_l_{       /_M'^         1-2        2.2ei-2 


(4)  +n(4)  =nP+(3ooo) 


1-6        3+^/b4+55  2Q^j^ 
^4  /  11 

For  2  =  5000,  this  gives: 

'(>  \  1-6 


2 


=  1.065    and    e  =  4.16kv.; 

hence, 

A  =  0.0338,  Efficency  96.62  per  cent.  Power  P= 20,800  kw. 

Method  of  Least  Squares. 

120.  An  interesting  and  very  important  appHcation  of  the 
theory  of  extremes  is  given  by  the  method  of  least  squares,  which 
is  used  to  calculate  the  most  accurate  values  of  the  constants 
of  functions  from  numerical  observations  which  are  more  numei- 
ous  than  the  constants. 

If  y=fix), (I) 


180  ENGINEERING  MATHEMATICS. 

is  a  function  having  the  constants  a,  b,  c  .  .  .  and  the  form  of 
tlie  function  (1)  is  known,  for  instance, 

y  =  a+bx+cx'^, (2) 

and  the  constants  a  b,  c  are  not  known,  but  the  numerical 
vahies  of  a  number  of  corresponding  values  of  x  and  y  are  given, 
for  instance,  by  experiment,  Xi,  J2,  -^3,  X4  .  .  .  and  2/1, 1/2, 2/3, 2/4  •  •  •  , 
then  from  these  corresponding  numerical  values  Xn  and  ?/„ 
the  constants  a,  b,  c  .  .  .  can  be  calculated,  if  the  numerical 
values,  that  is,  the  observed  points  of  the  curve,  are  sufficiently 
numerous. 

If  less  points  Xi  y\,  x^,  y2  ■  ■  ■  are  observed,  then  the  equa- 
tion (1)  has  constants,  obviously  these  constants  cannot  be 
calculated,  as  not  sufficient  data  are  available  therefor. 

If  the  number  of  observed  points  equals  the  number  of  con- 
stants, they  are  just  sufficient  to  calculate  the  constants.  For 
instance,  in  equation  (2),  if  three  corresponding  values  Xi,  yi] 
X2,  y2\  -^3,  2/3  arc  observed,  by  substituting  these  into  equation 
(2),  three  equations  are  obtained: 

y\  =  a-\-bx\  +cxi2;  1 

?/2  =  a+6.r2+c.r2^;  [ (3) 

?/3  =  03+&.r+CJ"3^,   J 

which  are  just  sufficient  for  the  calculation  of  the  three  constants 
a,  b,  c. 

Three  observations  would  therefore  be  sufficient  for  deter- 
mining three  constants,  if  the  observations  were  absolutely 
correct.  This,  however,  is  not  the  case,  but  the  observations 
always  contain  errors  of  observation,  that  is,  unavoidable  inac- 
curacies, and  constants  calculated  by  using  only  as  many 
observations  as  there  are  constants,  arc  not  very  accurate. 

Thus,  in  experimental  work,  always  more  observations 
are  made  than  just  necessary  for  the  determination  of  the 
constants,  for  the  purpose  of  getting  a  higher  accuracy.  Thus, 
for  instance,  in  astronomy,  for  the  calculation  of  the  orbit  of 
a  comet,  less  than  four  observations  are  theoretically  sufficient, 
but  if  possible  hundreds  are  taken,  to  get  a  greater  accuracy 
in  the  determination  of  the  constants  of  the  orbit. 


MAXIMA  AND  MINIMA.  181 

If,  then,  for  the  determination  of  the  constants  a,  h,  c  of 
equation  (2),  six  pairs  of  corresponding  values  of  x  and  y  were 
determined,  any  three  of  these  pairs  would  be  sufficient  to 
give  a,  h,  c,  as  seen  above,  but  using  different  sets  of  three 
observations,  would  not  give  the  same  values  of  a,  b,  c  (as  it 
should,  if  the  observations  were  absolutely  accurate),  but 
different  values,  and  none  of  these  values  would  have  as  high 
an  accuracy  as  can  be  reached  from  the  experimental  data, 
since  none  of  the  values  uses  all  observations. 

121.  If  y=A.v), (1) 

is  a  function  containing  the  constants  a,  h,  c  .  .  .,  which  arc  still 
unknown,  and  Xi,  yx]  Xo,  2/2;  2:3,  y^;  etc.,  arc  corresponding 
experimental  values,  then,  if  these  values  were  absolutely  cor- 
rect, and  the  correct  values  of  the  constants  a,h,  c  .  .  .  chosen, 
yi=f(xi)  would  be  true;    that  is, 

/(Ji)-yi=0;  1 

(5) 

/(j-2)-y2  =  0,etc.    J 

Due  to  the  errors  of  observation,  this  is  not  the  case,  but 
even  if  a,  6,  c  .  .  .  are  the  correct  values, 

yi^fixxU^ic: (6) 

that  is,  a  small  difference,  or  error,  exists,  thus 

/(xi)- 1/1  =  ^1;  1 

[ (7) 

/(j-2)-l/2  =  ^2,  etc.:  J 

If  instead  of  the  correct  values  of  the  constants,  a,  b,  c  .  .  ., 
other  values  were  chosen,  different  errors  ^1,  ^2  •  ■  •  would 
obviously  result. 

From  probability  calculation  it  follows,  that,  if  the  correct 
values  of  the  constants  a,  b,  c  .  .  .  are  chosen,  the  sum  of  the 
squares  of  the  errors, 

(5l2+0V+0V+ (g) 

is  less  than  for  any  other  value  of  the  constants  a,  b,  c  .  .  .;  that 
is,  it  is  a  minimum. 


182  ENGINEERING  MATHEMATICS. 

122.  The  problem  of  determining  the  constants  a,  h,  c.  .  ., 
thus  consists  in  finding  a  set  of  constants,  which  makes  the 
sum  of  the  squares  of  the  errors  3  a  minimum ;    that  is, 

0=  2I^2  =  i^inimum, (9) 

is  the  requirement,  which  gives  the  most  accurate  or  most 
probable  set  of  values  of  the  constants  a,  b,  c .  .  . 

Since  by  (7),  d=f(x)  —  y,  it  follows  from  (9)  as  the  condi- 
tion, which  gives  the  mj3st  probable  value  of  the  constants 
a,h,c.  .  .; 

2=  2lS/(j)  — 11/ p  =  minimum;      ....     (10) 

that  is,  the  least  sum  of  the  squares  of  the  errors  gives  the  most 
probable  value  of  the  constants  a,  h,  c .  .  . 

To  find  the  values  of  a,  6,  r  .  .  .,  which  fulfill  equation  (10), 
the  differential  quotients  of  (10)  are  equated  to  zero,  and  give 


da 


(11) 


This  gives  as  many  equations  as  there  are  constants  a,h,c  . . ., 
and  therefore  just  suffices  for  their  calculation,  and  the  values 
so  calculated  are  the  most  probable,  that  is,  the  most  accurate 
values. 

Where  extremely  high  accuracy  is  required,  as  for  instance 
in  astronomy  when  calculating  from  observations  extending 
over  a  few  months  only,  the  orbit  of  a  comet  which  possibly 
lasts  thousands  of  years,  the  method  of  least  squares  must  be 
used,  and  is  frequently  necessary  also  in  engineering,  to  get 
from  a  limited  number  of  observations  the  highest  accuracy 
of  the  constants. 

123.  As  instance,  the  method  of  least  squares  may  be  applied 
in  separating  from  the  observations  of  an  induction  motor, 
when  running  light,  the  component  losses,  as  friction,  hysteresis, 
etc. 


MAXIMA  AND  MINIMA. 


183 


In  a  440-volt  50-h.p.  induction  motor,  when  running  light, 
that  is,  without  load,  at  various  voltages,  let  the  terminal 
voltage  e,  the  current  input  i,  and  the  power  input  p  be  observed 
as  given  in  the  first  three  colunms  of  Table  I: 

Table    I 


e 

t 

p 

tV 

PO 

PO 

calc. 

J 

148 
220 
320 

8 
11 
19 

790 

020 

1500 

13 
24 
72 

780 

900 

1430 

746 

962 

1382 

+     32 
-  62 

+  48 

410 
440 
473 

23 
26 
29 

1920 
2220 
2450 

106 
135 
168 

1810 
2085 
2280 

1875 
2058 
2280 

-  35 

+  2.7 

0 

580 
640 

700 

43 
56 
75 

3700 
5000 
8000 

370 

627 

1125 

3330 
4370 
6875 

3080 
3600 
4150 

+  250 
+  770 
+  2725 

The  power  consumed  by  the  motor  while  running  light 
consists  of:  The  friction  loss,  which  can  be  assumed  as  con- 
stant, a;  the  hysteresis  loss,  which  is  proportional  to  the  1.6th 
power  of  the  magnetic  flux,  and  therefore  of  the  voltage,  be^-^; 
the  eddy  current  losses,  which  are  proportional  to  the  square 
of  the  magnetic  flux,  and  therefore  of  the  voltage,  ce^;  and  the  i-r 
loss  in  the  windings.    The  total  power  isj 


p  =  a  +  &e'  •'"'  +  ce^  +  ri^. 


(12) 


From  the  resistance  of  the  motor  windings,  r  =  0.2  ohm, 
and  the  observed  values  of  current  i,  the  ih  loss  is  calculated, 
and  tabulated  in  the  fourth  column  of  Table  I,  and  subtracted 
from  p,  leaving  as  the  total  mechanical  and  magnetic  losses  the 
values  of  po  given  in  the  fifth  column  of  the  table,  which  should 
be  expressed  by  the  equation  : 

p  =  a+6ei-c  +  ce2 (13) 

This  leaves  three  constants,  a,  6,  c,  to  be  calculated. 

Plotting  now  in  Fig.  59  with  values  of  e  as  abscissas,  the 
current  i  and  the  power  po  give  curves,  which  show  that  within 
the  voltage  range  of  the  test,  a  change  occurs  in  the  motor, 


184 


ENGINEERING  MATHEMATICS. 


as  indicated  by  the  abrupt  rise  of  current  and  of  power  beyond 
473  volts.  This  obviously  is  due  to  beginning  magnetic  satura- 
tion of  the  iron  structure.  Since  with  beginning  saturation 
a  change  of  the  magnetic  distribution  must  be  expected,  that 
is,  an  increase  of  the  magnetic  stray  field  and  thereby  increase 
of  eddy  current  losses,  it  is  probable  that  at  this  point  the  con- 


I 

n 

r 

/ 

/ 

,  ™ 

/ 

-6000 

y 

f 

/ 

-5000 

y 

/ 

/ 

9 

/ 

/ 

-4000 

y 

/ 

/ 

/ 

/ 

/ 

'/ 

/ 

-3000 

/ 

/ 

/ 

/ 

y 

J' 

y 

n 

-2000 

^^ 

y 

^ 

^ 

^ 

^ 

^ 

^ 

"  ^ 

^ 

y^ 

■^^ 

e= 

Volt 

> 

10 

^ 

1( 

0 

2( 

X) 

3( 

0 

4( 

w 

5( 

X) 

« 

0 

"( 

0 

Fig.  59.     Excitation  Power  of  Induction  Motor. 

stants  in  equation  (13)  change,  and  no  set  of  constants  can  be 
expected  to  represent  the  entire  range  of  observation.  For 
the  calculation  of  the  constants  in  (13),  thus  only  the  observa- 
tions below  the  range  of  magnetic  saturation  can  safely  be  used, 
that  is,  up  to  473  volts. 

From  equation  (13)  follows  as  the  error  of  an  individual 
observation  of  e  and  po: 


d  =  a  -\-he^-^  -\-  ce^  —  yo] 


(14) 


MAXIMA  AND  Ml  MIMA. 


185 


hence, 

thus : 


2=Sa2=S{a+6e^-«+ce2-;,op  =  mimmum,      (15) 
^  =  mia  +  he'-^'  +  ce^-  po]  =0: 


da 

dz 
db 


dz 


(16) 


and,  if  /i  is  the  number  of  observations  used  {n  =  6  in  this 
instance,  from  e  =  14S  to  e=-473),  this  gives  the  following 
equations : 

a:iei'>+6Ic-'^--+c^e^-6-^ei-«/)o  =  0;  '        .     .     (17) 
aSe2+6i:e3-''+cSe4-2e2p^j  =  o.  j 

Substituting  in  (17)  the  numerical  values  from  Table  I  gives, 

a  +  11.7  b  103  +  126  c  103  =  1550;  | 

a  +  14.6&  103  +  163  c  103  =  1830;  !■        .     .     (18) 

a +  15.1  b  103  +  170  c  103  =  1880;  J 


hence, 


a  =  540; 

6  =  32.5x10-3; 

c  =  5XlO-3, 


(19) 


and 


2^0  =  540+0.0325  6^1-6+0.005  e2 (20) 


The  values  of  po,  calculated  from  equation  (20),  are  given 
in  the  sixth  column  of  Table  I,  and  their  differences  from  the 
observed  values  in  the  last  colunm.  As  seen,  the  errors  are  in 
both  directions  from  the  calculated  values,  except  for  the  thi-ee 
highest  voltages,  in  which  the  observed  values  rapidly  increase 
beyond  the  calculated,  due  probably  to  the  appearance  of  a 


186  ENGINEERING  MATHEMATICS. 

loss  which  does  not  exist  at  lower  voltages — the  eddy  currents 
caused  by  the  magnetic  stray  field  of  saturation. 

This  rapid  divergency  of  the  observed  from  the  calculated 
values  at  high  voltages  shows  that  a  calculation  of  the  constants, 
based  on  all  observations,  would  have  led  to  wrong  values, 
and  demonstrates  the  necessity,  first,  to  critically  review  the 
series  of  observations,  before  using  them  for  deriving  constants, 
so  as  to  exclude  constant  errors  or  unidirectional  deviation.  It 
must  be  realized  that  the  method  of  least  squares  gives  the  most 
probable  value,  that  is,  the  most  accurate  results  derivable 
from  a  series  of  observations,  only  so  far  as  the  accidental 
errors  of  observations  are  concerned,  that  is,  such  errors  which 
follow  the  general  law  of  probability.  The  method  of  least 
squares,  however,  cannot  eliminate  constant  errors,  that  is, 
deviation  of  the  observations  which  have  the  tendency  to  be 
in  one  direction,  as  caused,  for  instance,  by  an  instrument  reading 
too  high,  or  too  low,  or  the  appearance  of  a  new  phenomenon 
in  a  part  of  the  observation,  as  an  additional  loss  in  above 
instance,  etc.  Against  such  constant  errors  only  a  critical 
review  and  study  of  the  method  and  the  means  of  observa- 
tion can  guard,  that  is,  judgment,  and  not  mathematical 
formalism. 

The  method  of  least  squares  gives  the  highest  accuracy 
available  with  a  given  number  of  observations,  but  is  frequently 
very  laborious,  especially  if  a  number  of  constants  are  to  be  cal- 
culated. It,  therefore,  is  mainly  employed  where  the  number  of 
observations  is  limited  and  cannot  be  increased  at  will;  but  where 
it  can  be  increased  by  taking  some  more  observations — as  is 
generally  the  case  with  experimental  engineering  investigations 
- — the  same  accuracy  is  usually  reached  in  a  shorter  time  by 
taking  a  few  more  observations  and  using  a  simpler  method 
of  calculation  of  the  constants,  as  the  2A-method  described  in 
paragraphs  153  to  157. 

Diophantic  Equations. 

1 23 A. — The  method  of  least  squares  deals  with  the  case, 
when  there  are  more  equations  than  unknown  quantities.  In 
this  case,  there  exists  no  set  of  values  of  the  unknown  quantities, 
which  exactly  satisfies  the  equations,  and  the  problem  is,  to  find 


MAXIMA  AND  MINIMA.  I860 

the  set  of  values,  which  most  nearly  satisfies  the  equations,  that 
is,  which  is  the  most  probable. 

Inversely,  sometimes  in  engineering  the  case  is  met,  when  there 
are  more  unknown  than  equations,  for  instance,  two  equations 
with  three  unknown  quantities.  Mathematically,  this  gives  not 
one,  but  an  infinite  series  of  sets  of  solutions  of  the  equations. 
Physically  however  in  such  a  case,  the  number  of  permissible 
solutions  may  be  limited  by  some  condition  outside  of  the  algebra 
of  equations.  Such  for  instance  often  is,  in  physics,  engineering, 
etc.,  the  condition  that  the  values  of  the  unknown  quantities 
must  be  positive  integer  numbers. 

Thus  an  engineering  problem  may  lead  to  two  equations  with 
three  unknown  quantities,  which  latter  are  limited  by  the  con- 
dition of  being  positive  and  integer,  or  similar  requirements, 
and  in  such  a  case,  the  number  of  solutions  of  the  equation  may 
be  finite,  although  there  are  more  equations  than  unknown 
quantities. 

For  instance: 

In  calculating  from  economic  consideration,  in  a  proposed 
hydroelectric  generating  station,  the  number  of  generators, 
exciters  and  step-up  transformers,  let: 

X  =  number  of  generators 

y  =  number  of  exciters 

z  =  number  of  transformers 

Suppose  now,  the  physical  and  economic  conditions  of  the 
installation  lead  us  to  the  equations: 

8x-\-3y  +  z  =  49  (1) 

2x +  y-\-Sz  =  21  (2) 

These  are  two  equations  with  three  unknown,  x,  y,  z;  these 
unknown  however  are  conditioned  by  the  physical  requirement, 
that  they  are  integer  positive  numbers. 

To  attempt  to  secure  a  third  equation  would  then  over  deter- 
mine the  problem,  and  give  either  wrong,  or  limited  results. 

Eliminating  z  from  (1)  and  (2),  gives: 

11a:  -\-  4y  =  as  (3) 


186&  ENGINEERING  MATHEMATICS. 

hence: 

3  —  3x 
since  y  must  be  an  integer  number,  — 7 —  must  also   be  an 

integer  number.     Call  this  u,  it  is: 


3-3x 

4       ="■ 

3  -4«       , 

-  u 

u 

~  3 

(5) 

since  x  must  be  an  integer  number,  5  must  also  be  an  integer 
number,  that  is: 

u  =  3y  6) 

hence,  substituted  into  (5),  (4)  and  (2): 


X  =  \  —  Aiv 
y  =  n-\-  llr 

z  =  2  -  V 


(7) 


(7)  thus  are  the  solutions  of  the  equations  (1)  (2),  where  v  is  any 
integer  number. 

As  seen,  mathematically,  there  are  an  infinite  number  of 
solutions. 

Substituting  now  for  v  integer  numbers: 

y=+2  +1  0-1-2 

X  =  -1  -3  +1        +5      +9 

2/=+ 35  +24  +13      +2      -9 

z  =       0  +1  +2        +3+4 

As  seen,  there  are  only  two  solutions,  for  v  =  0  and  v  =  —  \, 
which  give  for  x,  y,  and  z,  three  integer  positive  values,  and  which 
thus  satisfy  the  physical  restriction. 

r  =  0;  x  =  1,  y  =  13,  2  =  2  is  excluded  by  engineering  con- 
sideration, as  nobody  would  consider  thirteen  exciters  with  one 
generator,  and  thus  there  remains  only  one  applicable  solution: 


MAXIMA  AND  MINIMA.  186c 

X  =  5 
y  =  2 
z  =  3 

We  thus  have  here  the  case  of  two  equations  with  three  un- 
known quantities,  which  have  only  one  single  set  of  these  un- 
known quantities  satisfying  the  problem,  and  thus  give  a  definite 
solution,  though  mathematically  indefinite. 

This  type  of  equation  has  first  been  studied  by  Diophantes 
of  Alexandria. 


CHAPTER  V. 
METHODS  OF  APPROXIMATION. 

124.  The  investigation  even  of  apparently  simple  engineer- 
ing problems  frequently  leads  to  expre.ssions  which  are  so 
comphcated  as  to  make  the  numerical  calculations  of  a  series 
of  values  very  cumbersonme  and  almost  impossible  in  practical 
work.  Fortunately  in  many  such  cases  of  engineering  prob- 
lems, and  especially  in  the  field  of  electrical  engineering,  the 
different  quantities  which  enter  into  the  problem  are  of  very 
different  magnitude.  Many  apparently  complicated  expres- 
sions can  fiequently  be  gi'eatly  simplified,  to  such  an  extent  as 
to  permit  a  quick  calculation  of  numerical  values,  by  neglect- 
ing terms  which  are  so  small  that  their  omission  has  no  appre- 
ciable effect  on  the  accuracy  of  the  result;  that  is,  leaves  the 
result  correct  within  the  limits  of  accuracy  required  in  engineer- 
ing, which  usually,  depending  on  the  nature  of  the  problem, 
is  not  greater  than  from  0.1  per  cent  to  1  per  cent. 

Thus,  for  instance,  the  voltage  consumed  by  the  resistance 
of  an  alternating-current  transformer  is  at  full  load  current 
only  a  small  fraction  of  the  supply  voltage,  and  the  exciting 
.current  of  the  transformer  is  only  a  small  fraction  of  the  full 
load  current,  and,  therefore,  the  voltage  consumed  by  the 
exciting  current  in  the  resistance  of  the  transformer  is  only 
a  small  fraction  of  a  small  fraction  of  the  supply  voltage,  hence, 
it  is  negligible  in  most  cases,  and  the  transformer  equations  are 
greatly  simplified  iDy  omitting  it.  The  power  loss  in  a  large 
generator  or  motor  is  a  small  fraction  of  the  input  or  output, 
the  drop  of  speed  at  load  in  an  induction  motor  or  direct- 
current  shunt  motor  is  a  small  fraction  of  the  speed,  etc.,  and 
the  square  of  this  fraction  can  in  most  cases  be  neglected,  and 
the  expression  simplified  thereby. 

Frequently,  therefore,  in  engineering  expressions  con- 
taining  small   quantities,   the   products,   squares   and   higher 

187 


188  ENGINEERING  MATHEMATICS. 

powers  of  such  quantities  may  be  dropped  and  the  expression 
thereby  simphfied;  or,  if  the  quantities  are  not  quite  as  small 
as  to  permit  the  neglect  of  their  squares,  or  where  a  high 
accuracy  is  required,  the  first  and  second  powers  may  be  retained 
and  only  the  cubes  and  higher  powers  dropped. 

The  most  common  method  of  procedure  is,  to  resolve  the 
expression  into  an  infinite  series  of  successive  powers  of  the 
small  quantity,  and  then  retain  of  this  series  only  th(^  first 
term,  or  only  the  first  two  or  three  terms,  etc.,  depending  on  the 
smallness  of  the  quantity  and  the  required  accuracy. 

125.  The  forms  most  frequently  used  in  the  reduction  of 
expressions  containing  small  quantities  are  multiplication  and 
division,  the  binomial  series,  the  exponential  and  the  logarithmic 
series,  the  sine  and  the  cosine  series,  etc. 

Denoting  a  small  quantity  by  s,  and  where  seA'^eral  occur, 
by  si,  S2,  S3  .  .  .  the  following  expression  holds: 

(1  ±Sl)(l±S2)=l±Sl±S2±SlS2, 

and,  since  Si52  is  small  compared  with  the  small  quantities 
si  and  S2,  or,  as  usually  expressed,  S\S2  is  a  small  quantity  of 
higher  order  (in  this  case  of  second  order),  it  may  be  neglected, 
and  the  expression  written : 

(l±-Sl)(l±S2)  =  l±Si±S2 (1) 

This  is  one  of  the  most  useful  simplifications :  the  multiplica- 
tion of  terms  containing  small  quantities  is  replaced  by  the 
simple  addition  of  the  small  quantities. 

If  the  small  quantities  Si  and  S2  are  not  added  (or  subtracted) 
to  1,  but  to  other  finite,  that  is,  not  small  quantities  a  and  h, 
a  and  b  c£in  be  taken  out  as  factors,  thus, 

(a±Si)(6±S2Wa6A±^Vl±|')=a6('l±^-±|V  .     (2) 

where  —  and  j-  nmst  be  small  quantities. 

As  seen,  in  this  case,  si  and  So  need  not  necessaril}'  be  abso- 
luloly  small  quantities,  but  may  be  quite  large,  provided  that 
a  and  b  are  still  larger  in  magnitude;  that  is,  si  must  be  small 
compared  with  a,  and  S2  small  compared  with  b.    For  instance, 


METHODS  OF  APPROXIMATION.  189 

in  astronomical  calculations  the  mass  of  the  earth  i^which 
absolutely  can  certainly  not  be  considered  a  small  quantity) 
is  neglected  as  small  quantity  compared  with  the  mass  of  the 
sun.  Also  in  the  effect  ol  a  lightning  stroke  on  a  primary 
distribution  circuit,  the  normal  line  voltage  of  2200  may  be 
neglected  as  small  compared  with  the  voltage  impressed  by 
lightning,  etc. 

126.  Example.  In  a  direct-current  shunt  motor,  the  im- 
pressed voltage  is  eo  =  125  volts;  the  armature  resistance  is 
ro  =  0.02  ohm;  the  field  resistance  is  ri  =  50  ohms;  the  power 
consumed  by  friction  is  p/=-300  watts,  and  the  pow^r  consumed 
by  iron  loss  is  pi  =  iOO  watts.  What  is  the  power  output  of 
the  motor  at  io  =  50, 100  and  150  amperes  input? 

The  power  produced  at  the  armature  conductors  is  the 
product  of  the  voltage  e  generated  in  the  armature  conductors, 
and  the  current  i  through  the  armature,  and  the  power  output 
at  the  motor  pulley  is, 

P-ei-pf-pi (3) 


ent  m  ine  moior  neia  is 
therefore  is, 


The  current  in  the  motor  field  is  — ,  and  the  armature  current 

ri 


i  =  ^o--,        (4) 

ri 


Co  . 


where  —  is  a  small  quantity,  compared  with  I'o. 

The  voltage  consumed  by  the  armature  resistance  is  roi, 
and  the  voltage  generated  in  the  motor  armature  thus  is: 

e  =  eo-roi, (5) 

where  TqI  is  a  small  quantity  compared  with  e^^. 
Substituting  herein  for  i  the  value  (4)  gives, 

e  =  €o-rJiu-:^j (6) 


Since  the  second  term  of   (6)  is  small  compared  with  eo, 

€0 
and  in  this  second  term,  the  second  term  —  is  small  com- 

pared  with  io,  it  can  be  neglected  as  a  small  term  of  higher 


190  ENGINEERING  MATHEMATICS. 

order;    that  is,  as  small  compared  with  a    small  term,  and 
expression  (6)  simplified  to 

e  =  eo-roio (7) 

Substituting  (4)  and  (7)  into  (3)  gives, 

p  =  (eo  -  raio)  f  lo  -f)-  Vf-  Vi 

Expression  (8)  contains  a  product  of  two  terms  with  small 
quantities,  which  can  be  multiplied  by  equation  (1),  and  thereby 
gives, 

.  /,     roio      eo  \ 

p  =  eoio(l — —--Tj-Pf-Pi 
\        eo      r\io/ 

=  eoio-roiV--r — pf-pi (9) 

Substituting  the  numerical  values  gives, 

p  =  125io-  0.02io2-  562.5-  300-  400 
=  125?o - 0.02?o2 - 1260  approximately ; 

thus,  for  2o  =  50,  100,  and  150  amperes;    p  =  4940,  11,040,  and 
17,040  watts  respectively. 

127.  Expressions  containing  a  small  quantity  in  the  denom- 
inator are  frequently  simplified  by  bringing  the  small  quantity 
in  the  numerator,  by  division  as  discussed  in  Chapter  II  para- 
graph 39,  that  is,  by  the  series, 

-J— =  l:fX+.r2qFi-3+i-^iFi5+ (10) 

l±.r 

which  series,  if  x  is  a  small  quantity  s,  can  be  approximated 
by: 


'    =1-.: 


1+s 

1 


\  —  s 


(11) 


METHODS  OF  APPROXIMATION. 


191 


or,  where  a  greater  accuracy  is  required, 

1 


1+s 
_1_ 


=  l-s+s2; 

=  1+S+S2. 


(12) 


By  the  same  expressions  (11)  and  (12)  a  small  quantity 
contained  in  the  numerator  may  be  brought  into  the  denominator 
where  this  is  more  convenient,  thus : 


l+.s  = 


1 


l  —  S  =  -:r--r~',   etc. 

1+s' 


(13) 


More  generally  then,  an  expression  like  ,  where  s  is 

small  compared  with  a,  may  be  simpUfied  by  approximation  to 
the  form, 


a±s 


«('-«-)  " 


^(..^ 


(14) 


or,  where  a  greater  exactness  is  required,  by  taking  in  the  second 
term, 


_b b/ 

a±s     a\ 


a     a- 


(15) 


128.  Example.  What  is  the  current  input  to  an  induction 
motor,  at  impressed  voltage  eo  and  slip  s  (given  as  fraction  ot 
synchronous  speed)  if  ro  +  yxo  is  the  impedance  of  the  primary 
circuit  of  the  motor,  and  r\  ^-jx\  the  impedance  of  the  secondary 
circuit  of  the  motor  at  full  frequency,  and  the  exciting  current 
of  the  motor  is  neglected;  assuming  s  to  be  a  small  quantity; 
that  is,  the  motor  running  at  full  speed? 

Let  E  be  the  e.m.f.  generated  by  the  mutual  magnetic  flux, 
that  is,  the  magnetic  flux  which  interlinks  with  primary  and 
with  secondary  cu'cuit,  in  the  primary  circuit.  Since  the  fre- 
quency of  the  secondary  circuit  is  the  fraction  s  of  the  frequency 


192  ENGINEERING  MATHEMATICS. 

of  the  primary  cii'cuit,  the  generated  e.iii.f.  o^  the  secondary 
circuit  is  sE. 

Since  Z\  is  the  reactance  of  the  secondary  circuit  at  full 
frequency,  at  the  fraction  s  of  full  frequency  the  reactance 
of  the  secondary  circuit  is  sx\,  and  the  impedance  of  the  sec- 
ondary circuit  at  slip  s,  therefore,  is  r\+jsx\]  hence  the 
secondary  current  is, 

•     ri  +  j.s.ri 

If  the  exciting  current  is  neglected,  the  primary  current 
equals  the  secondary  current  (assuming  the  secondary  of  the 
same  number  of  turns  as  the  primary,  or  reduced  to  the  same 
number  of  turns) ;  hence,  the  current  input  into  the  motor  is 

I^^ (16) 

The  second  term  in  the  denominator  is  small  compared 
with  the  first  term,  and  the  expression  (16)  thus  can  be 
approximated  by 

'^    ^^^-/?i) (17) 


•    .(:.;^')    -       - 

The  voltage  E  generated  in  the  primary  circuit  equals  the 
impressed  voltage  eo,  minus  the  voltage  consumed  by  the 
current  /  in  the  primary  impedance;  ro+jj^o  thus  is 

?  =  fo-/(ro+j>o) (18) 

Substituting  (17)  into  (18)  gives 

E  =  e„-^(ro+/xo)(l-./^) (19) 

In  expression  (19),  the  second  term  on  the  right-hand  side, 
which  is  the  impedance  drop  in  the  primary  circuit,  is  small 

1  — /-^ 

of  this  small  term,  the  small  term  i —  can  thus  be  neglected 

•"  ri  ° 


METHODS  OF  APPROXIMATION.  193 

as  a  small  term  of  higher  order,  and  equation  (19)  abbreviated 
to 

sE 

E  =  eo-  —  (ro+jxo) (20) 

From  (20)  it  follows  that 

and  by  (13), 

E  =  Coll-~(ro+jxo)] (21) 

Substituting  (21)  into  (17)  gives 

and  by  (1), 

,    sec f        ..s.r,      s       ,.    ,  1 

=  f!2    l_,r=_y,,fi±f!| (22) 

If  then,   loo-^io—jio'  is    the    exciting    current,  the    total 
current  input  into  the  motor  is,  approximately, 

/o=(+/oo 

seo  { ^       To     .  Xo+X\] 


ri  [         ri     "       ri 

129.  One  of  the  most  important    expressions  used  for  the 
reduction  of  small  terms  is  the  binomial  series: 

n(n—l^  „    n(n— l)(n  — 2)    „ 
(l±.r)"  =  l±nx+-^5 — x^± ^^ -x^ 

n{n-l)(n-2)(n-S)  ^ 
+- -[4— ^  ± .  . .      (24) 

If  .T  is  a  small  term  s,  this  gives  the  approximation, 

a±sy  =  l±ns; (25) 


194  ENGINEERING  MATHEMATICS. 

or,  using  the  second  term  also,  it  gives 

(l±s)-==l±ns+'^—\^ (26) 

In  a  more  general  form,  this  expression  gives 

1±-)   =a"ll±— );etc.      .     .     (27) 

By  the  binomial,  higher  powers  of  terms  containing  5?mall 
quantities,  and,  assuming  n  as  a  fraction,  roots  containing 
small  quantities,  can  be  eliminated;  for  instance. 


(a±s)" 

a"[l± 
a 


1 _J1_A     ^\""_J_A     ns\ 

sY     a"\      a)         a"\        a/' 


-n-=-  =  (a±s)    n=a    "(l±-)     "=— =(lq:  — ); 

m 

«/7 T-      /        N-      -/.      ^\^     n/ — /,     rns\ 

v(a±s)'»  =  (a±s)"  =a"(  1  ±- I    =Va^[l± — );  etc. 

One  of  the  most  common  uses  of  the  binomial  series  is  for 
the  elimination  of  squares  and  square  roots,  and  very  fre- 
quently it  can  be  conveniently  applied  in  mere  numerical  calcu- 
lations; as,  for  instance, 

(201)2  =  2002(1  +^jj  =  40,000(l  +~^  =40,400; 
29.92  =  302(l-4)^900(l-4)=000-6  =  894; 


V9a8  =  10\/l-0.02  =  10(l-0.02)2  =10(1 -0.01)  =9.99; 

vrOi  ^  (1+0.03)  1/2  =roT5^^-^^'^'  ^*^- 


METHODS  OF  APPROXIMATION.  195 

130.  Example  i.  If  r  is  the  resistance,  x  the  reactance  of  an 
alternating-current  circuit  with  impressed  voltage  e,  the 
current  is 

c 

2=- 


Vf'^+x- 

If  the  reactance  x  is  small  compared  with  the  resistance  r, 
as  is  the  case  in  an  incandescent  lamp  circuit,  then, 

Vr2+x2         I       Yj.\2     r  [        \r 


-'-11 


■Mr) 

]_/x 
2\r 


If  the  resistance  is  small  compared  with  the  reactance,  as 
is  the  case  in  a  reactive  coil,  then, 


1 


._       e       _  e  e  (       /^\~^     ^ 

Vr2+x2  r      7r'Y~x[        \xj  i 


x^ll+{~ 

.X 


ef.     1/r 


-P-2WI "•''' 

Example  2.  How  does  the  short-circuit  current  of  an 
alternator  vary  with  the  speed,  at  constant  field  excitation? 

When  an  alternator  is  short  circuited,  the  total  voltage 
generated  in  its  armature  is  consumed  by  the  resistance  and  the 
synchronous  reactance  of  the  armature. 

The  voltage  generated  in  the  armature  at  constant  field 
excitation  is  proportional  to  its  speed.  Therefore,  if  eo  is  the 
voltage  generated  in  the  armature  at  some  given  speed  So, 
for  instance,  the  rated  speed  of  the  machine,  the  voltage 
generated  at  any  other  speed  S  is 

S 
00 


196  ENGINEERING  MATHEMATICS. 

or,  if  for  convenience,  the  fraction  -tt  if>  denoted  bv  a,  then 
a  =  -^     and     e  =  aeo, 

where  a  is  the  ratio  of  the  actual  speed,  to  that  speed  at  which 
the  generated  voltage  is  eo. 

If  r  is  the  resistance  of  the  alternator  armature,  Xo  the 
synchronous  reactance  at  speed  So,  the  synchronous  reactance 
at  speed  S  is  x  =  axo,  and  the  current  at  short  circuit  then  is 

t._i==-^fl= f29) 

V  r^  +  j;2     \  /'^  +  a'^xo^ 

Usually  r  and  xo  are  of  such  magnitude  that  r  consumes 
at  full  load  about  1  per  cent  or  less  of  the  generated  voltage, 
while  the  reactance  voltage  of  xo  is  of  the  magnitude  of  from 
20  to  50  per  cent.  Thus  r  is  small  compared  with  xq,  and  if 
a  is  not  very  small,  equation  (29)  can  be  approximated  by 


aeo eo\    _^(_^_\ 

I       /  r  \2 ~ Xo\       2  \axo/ 


axo 

\axo 


(30) 


Then  if  xo  =  20r,  the  following  relations  exist: 
o=        0.2  0.5  1.0  2.0 

1  =  -X0.9688      0.995      0.99875      0.99969 

.To 

That  is,  the  short-circuit  current  of  an  alternator  is  practi- 
cally constant  independent  of  the  speed,  and  begins  to  decrease 
only  at  very  low  speeds. 

131.  Exponential  functions,  logarithms,  and  trigonometric 
functions  are  the  ones  trequc^ntly  met  in  electrical  engineering. 

The  exponential  function  is  defined  by  the  series, 

/v«2  j^O  j*4  y^S 


METHODS  OF  APPROXIMATION.  197 

and,  if  X  is  a  small  quantity,  s,  the  exponential  function,  may 
be  approximated  by  the  equation, 

c±«  =  l-i-s; (32) 

or,  by  the  more  general  equation, 

£±"''  =  l±as; (33) 

and,  if  a  greater  accuracy  is  required,  the  second  term  may 
be  included,  thus, 


£±«=l±s4-^, (34) 


and  then 


2o2 


a^s 


£±as  =  l-ta,s+-— (35) 

fdx 
The  logarithm  is  defined  by  log£  J  ==   I — ;  hence. 


log,fl±.r)=±J^|;^. 


Resolving  ■T-r~  ^^^^^'  ^  scries,  by  (10),  and  then  integrating, 
gives 

log£(l±i-)=±  (  {lTx+x^:^x^  +  ...)dx 

.r2     x^     x^     ;i5 
■=±x-^±j-j±^- (36) 

This  logarithmic  series  (36)  leads  to  the  ai)i)r()ximation, 

logai±s)=±s; (37) 

or,  including  the  secontl  term,  it  gives 

\oge{l±s)=±s-s^, (38) 

and  the  more  general  expression  is,  respectively, 

loge  (a±s)  =  log  fl^l  ±^- j  =log  a  +log  (l  ±^)  =log  a±^,      (39) 


198 


ENGINEERING  MATHEMATICS 


and,  more  accurately, 


loge  (a±6')  =  loga± 


s     s^ 
a    o? 


(40) 


Since  logio  A7'  =  logio  fXlogc  iV  =  0.4343  logc  A/",  equations  (^39) 
and  (40)  may  be  written  thus, 


logio(l±s)=±  0.4343s; 
logio  (a±s)  =logio  a  ±0.4343- 


(41) 


132.  The   trigonometric   functions    are  represented   by   the 
infinite  series : 


x^    y^    ^ 
cos.r  =  l-j2+|4-je+.- 

x^    ofi    x^ 
sm  2-  =  x— 777+TF  — pr  +. 
E     1^     li 


(42) 


which  when  s  is  a  small  quantity,  may  be  approximated  by 

coss  =  ]     and    sin  .s  =  s;        .     .     .     .     (43) 
or,  they  may  be  represented  in  closer  approximation  by 

S2 


cos  s  =  1  — 


2' 


sm  5  =  s 


or,  by  the  more  general  expressions, 


(44) 


cos  as  =  1      and     cos  as  =  l 


2o2 


a^8 


0   ' 


sin  as  =  as    and     sin  a.s  = 


"^  l-irj- 


(45) 


133.  Other  functions  containing  small  terms  may  frequently 
be  approximated  by  Taylor's  series,  or  its  special  case, 
MacLaurin's  series. 

MacLaurin's  series  is  written  thus : 


fix)  =/(0)  +xf(0)  ^p'\0)  +^r'(0)  +. 


(46) 


METHODS  OF  APPROXIMATION.  199 

where  /',  /",  f",  etc.,  are  respectively  the  first,  second,  third, 
etc.,  differential  quotient  of/;  hence. 


fias)=m+asf{0). 
Taylor's  series  is  written  thus. 


(47^ 


f{b+x)  =J\b)  +xf\b)  +|r  (6)  +p"'(b)  +...,     .     (48) 


(49) 


and  leads  to  the  approximations : 

f{h±s)=f{b)±sf'{b); 

f(b±as)=f{b)±asf{b).    J 

Many  of  the  previously  discussed  approximations  can  be 
considered  as  special  cases  of  (47)  and  (49). 

134.  As  seen  in  the  preceding,  convenient  equations  for  the 
approximation  of  expressions  containing  small  terms  are 
derived  from  various  infinite  series,  which  are  summarized 
below : 

,,  ,    N      -,             n(n-l)  „    n(n— l)(n-2)  , 
(l±.-r)"  =  l±nx+      .^      x^±- j^^ -V  +  . 

'j'S        '1*3        'j«4 

e±^  =  l±.r  +  p±j3+i^±...; 


loge  (1  ±x)  =  ±.r-^  ±  3--  J  ±.  .  .  ; 

^     x^    x'*    x^ 
cosa:  =  l-p+p-j^+...; 

nrS         v-S         Y»7 

sin  x  =  j— 777+^  — ^  +  .  .  .  ; 
Ax)  =/(0)  +xf(0)  +|>(0)  +^/-'(0)  +.  , 
/(6  ±.t)  =f{b)  ±xf\b)  +tf"{b)  ±trib)  +  . 


\    (.^0) 


200 


ENGINEERING  MATHEMATICS. 


The  first  approximations,  derived  by  neglecting  all  higher 
terms  but  the  first  power  of  the  small  quantity  x  =  s  in  these 
series,  are : 


1  ±s 

log^n  ±.s^=  ±.s; 
cog  s  =  1 ; 

sin  s  =  s; 

/(•^)=-/(0)+.srfO); 
f(h±s)=f(h)±sr{h); 


[+^']; 


nin—l)  „"1 

4]- 

.S2  "1 


(51) 


and,  in  addition  hereto  is  to  be  remembered  the  multiplication 
rule, 

(l±.Sl)(l±.S2)  =  l±.Sl±S2;         [±SiS2].         .      .       (52) 

135.  The  accuracy  of  the  approximation  can  be  estimated 
by  calculating  the  next  term  beyond  that  which  is  used. 
This  term  is  given  in  brackets  in  the  above  equations  (50) 
and  (51). 

Thus,  when  calculating  a  series  of  numerical  values  by 
approximation,  for  the  one  value,  for  which,  as  seen  by  the 
nature  of  the  problem,  the  approximation  is  least  close,  the 
next  term  is  calculated,  and  if  this  is  less  than  the  permissible 
limits  of  accuracy,  the  approximation  is  satisfactory. 

For  instance,  in  Example  2  of  paragraph  130,  the  approxi- 
mate value  of  the  short-circuit  current  was  found  in  (30),  as 


.ro 


-(--Tl- 


METHODS  OF  APPROXIMATION.  201 

The  next  term  in  the  parenthesis  of  equation  (30),  by  the 
binomial,  would  have  been   -\ — s^;   substituting  n=— ^; 


s  = 


r  Y  3  /  r  \  1 

— )  ,  the  next  becomes  +^(  —  I  .    The  smaller  the  a,  the 
a.Vo/  o  \axo/ 

less  exact  is  the  approximation. 

The  smallest  value  of  a,  considered  in  paragraph  130,  was 

3  /  r  V 
a  =  0.2.     For  xo  =  20r,  this  gives    +^(  — )  =0.00146,  as  the 


^axo/ 
value  of  the  first  neglected  term,  and  in  the  accuracy  of  the 

result  this  is  of  the  magnitude  of  -XO.00146,  out  of  —  X  0.9688, 

Xj  '  Xo  ' 

the  value  given  in  paragraph  130;   that  is,  the  approximation 

gives  the  result  correctlv  within    '  ..nQ,.-  =0.0015  or  within  one- 

sixth  of  one  per  cent,  which  is  sufficiently  close  for  all  engineer- 
ing purposes,  and  with  larger  a  the  values  are  still  closer 
approximations. 

136.  It  is  interesting  to  note  the  different  expressions, 
which  are  approximated  by  (1 +s)  and  by  (1  — s).  Some  of 
them  are  given  in  the  following : 

l-s  = 
_1_ 

l+s' 

s 

1-- 

1--T 


l+s  = 

1 

1- 

s' 

/       A** 
\      n/   ' 

(       sY 

V  ^2)  ■' 

1 

('■ 

nJ 

( 

111 
n- 

\^ 

n—m 

(-^)' 


202 


ENGINEERING  MATHEMATICS. 


1 


Vl-2s' 

/f+s 
\l-s' 

1 


ns 


l+nlog.l^l+- 
l-nlog.(l-^ 


1-log, 

etc. 
1  +sin  s; 


El. 
\fl+s'. 


Vl-2s; 
1 

VlT2s' 


EI. 

v^l  — ns; 
1 


\l  +  (ji-m 


n—7?i)s 
etc. 

; 

l+log£(l-s); 
l-logeO+.s); 


l+nlog.(^l--j; 
l-nlog.(l+^-); 


l+lo 


1-log, 


etc. 


1  — sin  s; 


ll-nsin  — ; 
n' 


1  —  ?i  pin  — ; 
n 


METHODS  OF  APPROXIMATION. 


203 


.      1    • 

1  +—  sin  ns\ 

n  ' 


cos  V  — 2s; 
etc. 


1 sin  ns'. 

n  ' 

cos  \/2s; 
etc. 


137.  As  an  example  may  be  considered  the  reduction  to  its 
simplest  form,  of  the  expression : 


F= 


2£i  l2g 

Va  \'(a  +  .si)3j4  — sin  6s2(  •i'aso  qq^2    Ll. 

\  a 


-3S2 


.a+2s0|l-aloge^^-p|-|va-2.9i 


then, 


^0^+^=  (a  +  sO^/^^-a*"!  1  +^]      =aJ(l  + 


^■A 


3/4 


a/' 


4— sin  6.?2  =  4   I 


^sin6s2J=-4(^l-^S2J; 


2.S1 


Sl 


£«  =1+2-; 


Sl 


Sl 


cos2.  — =    1 — ~     =1-2-; 
\  a      \       a/  a' 

,-3sj_1  _:»?„. 
;  —  J        Oc2 1 


,+2.,  =  a(l+2^); 


r—  A--  , 


=  1  — a  logcf  1- 


=  1+S2; 


204  ENGINEERING  MATHEMATICS. 

hence, 
^    ,..Xa3/.(l4^)x4(l-|s.)xa./^x(l+2^)(l-2li) 

(1  -Ssj)  X«(l  +2^)(1  +si)  Xa'/^A -ilj 

4a3/<l+|!l-|.,,  +  2^-2il) 
\       4  a      2  a       a  / 

a3/2(l-3s2+2-"-+S2--) 
\  a  a/ 


=  4ll-i^+.^ 


a     2a/ 


138.  As  further  example  may  be  considered  the  equations 
of  an  alternating-current  electric  circuit,  containing  distributed 
resistance,  inductance,  capacity,  and  shunted  conductance,  for 
instance,  a  long-distance  transmission  line  or  an  underground 
high-potential  cable. 

Equations  of  the  Transmission  Line. 

Let  I  be  the  distance  along  the  line,  from  some  starting 
point;  E,  the  voltage;  /,  the  current  at  point  /,  expressed  as 
vector  quantities  or  general  numbers;  Zo  =  ro+jxo,  the  line 
impedance  per  unit  length  (for  instance,  per  mile);  Yo=go+jho 
=  line  admittance,  shunted,  per  unit  length;  that  is,  tq  is  the 
ohmic  effective  resistance;  xo,  the  self-inductive  reactance; 
60,  the  condensive  susceptance,  that  is,  wattless  charging 
current  divided  by  volts,  and  go  =  energy  component  of  admit- 
tance, that  is,  energy  component  of  charging  current,  divided 
by  volts,  per  unit  length,  as,  per  mile. 

Considering  a  line  element  dl,  the  voltage,  dE,  consumed 
by  the  impedance  is  Z^Idl,  and  the  current,  dl,  consumed  by 
the  admittance  is  YoEdl]  hence,  the  following  relations  may  be 
written : 

Tr'-^i- (1) 

f-^'"? (2) 


METHODS  OF  APPROXIMATION. 
Differentiating  (1),  and  substituting  (2)  therein  gives 


dP 


-ZoYoE, 


and  from  (1)  it  follows  that, 

\_dE 

Zo  dl  ' 

Equation  (3)  is  integrated  by 

E  =  AeBl, 
and  (5)  sub>tituted  in  (3)  gives 


205 


(3) 


C4) 


B=±VZoYo;     .    . 
hence,  from  (5)  and  (4),  it  follows 


(5) 

(6) 

(7) 
(8) 


Next  assume 

l  =  lo,      the  entire  length  of  line; 
Z  =  Zo-^o,  the  total  line  impedance;     \,    .     .     .     (9) 
and  F  =  ^o^o,  the  total  line  admittance; 

then,  substituting  (9)  into  (7)  and  (8),  the  following  expressions 
are  obtained : 


/i 


{Ais  +  '^^y-Aoe-^^Y], 


(10) 


as  the  voltage  and  current  at  the  generator  end  of  the  line. 

139.  If  now  Eo  and  /o  respectively  are  the  current  and 
voltage  at  the  step-down  end  of  the  line,  for  /  =  0,  by  sub- 
stituting 1  =  0  into  (7)  and  (8), 

A,+A.2=Eo; 


Ax-A2  =  l 


O^y. 


(11) 


206 


ENGINEERING  MATHEMATICS. 


Substituting  in  (10)  for  the  exponential  function,  the  series, 


, ZY    ZYx^ZY     Z^Y^    Z^Y^VZY 

/      ZY    Z2y2\  /      ZY  ,  Z272\ 


(12) 


and  arranging  by   (Ai+M)  and   (.41-42),  and  substituting 
herefor  the  expressions  (11),  gives 


^      ^   ,        ZY     Z^Y^] 


^,     ,       ZY    Z^Y^] 

+ZI0  \^+-Q-+^20']' 


^      ,    f       ZY    Z^Y'-]      ,,^   \^     ZY    Z^Y^] 


(13) 


AVhen  l  =  —lo,  that  is,  for  ^0  and  Iq  at  the  generator  side,  and 
El  and  /i  at  the  step-down  side  of  the  line,  the  sign  of  the 
second  term  of  equations  (13)  merely  reverses. 

140.  From  the  foregoing,  it  follows  that,  if  Z  is  the  total 
impedance;  Y,  the  total  shunted  admittance  of  a  transmission 
line,  'Eo  and  /o,  the  voltage  and  current  at  one  end;  Ex  and  [1, 
the  voltage  and  current  at  the  other  end  of  the  transmission 
line;  then. 


.^      ^   f       ZY    Z^Y^]      ^,  [,     ZY    Z^Y^] 
^i  =  ^o|l+^+-,:^|±Z/o|l+^+-^|: 

,      ,    r       ZY    Z^Y^]      ^^^   f^     ZY    Z^Y^] 


24 


G 


120  J  ' 


(14) 


where  the  plus  sign  applies  if  Eq,  Iq  is  the  step-down  end, 
the  minus  sign,  if  £"0,  /o  is  the  step-up  end  of  the  transmission 
lino. 

In  practically  all  cases,  the  quadratic  term  can  be  neglected, 
and  the  equations  simplified,  thus, 
r 


(15) 


7V]  {        ZY 

£^i=^oil+^j±  ^(o|l+-^ 

r       ZY}  f       ZY] 

and  the  error  made  hereby  is  of  the  magnitude  of  less  than 


2, ,2 


24 


METHODS  OF  APPROXIMATION.  207 

Except  in  the  case  of  very  long  lines,  the  second   term  of 
the  second  term  can  also  usually  be  neglected,  which  givcb 


(16) 


and  the  error  made  hereby  is  of  the  magnitude  of  less  than  — 

6 

of  the  Une  impedance  voltage  and  line  charging  current. 

141.  Example.  Assume  200  miles  of  60-cycle  line,  on  non- 
inductive  load  of  60  =  100,000  volts;  and  io  =  100  amperes. 
The  line  constants,  as  taken  from  tables  are  ^  =  104  +  140/  ohms 
and  y  = +0.0013]  ohms;  hence, 

Zr=-(0. 182 -0.186/); 

El  =  100000  (1  -  0.091 +0.068/)  + 100  (104  + 140/) 
=  101400+20800/,  in  volts ; 

/i  =  100(1 -0.091  +  0.068/)+0.0013/Xl00000 
=  91  +  136.8/,  in  amperes. 

.    2y     0.174X0.0013     0.226 
The  error  is  -^  = 7; =  — ^ —  =  0.038. 

DO  6 

In  El,  the  neglect  of  the  second  term  of  2:/o  =  17,400,  gives 
an  error  of  0.038x17,400  =  660  volts  =  0.6  per  cent. 

In  /i,  the  neglect  of  the  second  term  of  ^£"0  =  130,  gives  an 
error  of  0.038x130  =  5  amperes  =3  per  cent. 

Although  the  charging  current  of  the  line  is  130  per  cent 
of  output  current,  the  error  in  the  current  is  only  3  per  cent. 

Using  the  equations  (15),  which  are  nearly  as  simple,  brings 

zh/^    0.2262 
uhe  error  down  to  -ttt""""^! —  =  0.0021,  or  less  than  one-quarter 

per  cent. 

Hence,  only  in  extreme  cases  the  equations  (14)  need  to  be 

used.     Their  error  would  be  less  than  ^^7;j^  =  3.6x10~6,  or  one 

fchree-thousandth  per  cent. 


208  ENGINEERING  MATHEMATICS. 

The  accuracy  of  the  preceding  approximation  can  be  esti- 
mated by  considering  the  physical  meaning  of  Z  and  Y\  Z 
is  the  hne  impedance;    hence  Zl  the  impedance  voltage,  and 

zi 

w  =  -p,  the  impedance  voltage  of  the  line,  as  fraction  of  total 

voltage;    Y  is  the  shunted  admittance;  hence  YE  the  charging 

YE  . 

current,  and  v  =  —j-,  the  charging  current  of  the  line,  as  fraction 

of  total  current. 

Multiplying  gives  uv  =  ZY;  that  is,  the  constant  ZY  is  the 
product  of  impedance  voltage  and  charging  current,  expressed 
as  fractions  of  full  voltage  and  full  current,  respectively.  In 
any  economically  feasible  power  transmission,  irrespective  of 
its  length,  both  of  these  fractions,  and  especially  the  first, 
must  be  relatively  small,  and  their  product  therefore  is  a  small 
quantity,  and  its  higher  powers  negligible. 

In  any  economically  feasible  constant  potential  transmission 
line  the  preceding  approximations  are  therefore  permissible. 


Approximation  by  Chain  Fraction. 

141A. — A  convenient  method  of  approximating  numerical 
values  is  often  afforded  by  the  chain  fraction.  A  chain  fraction 
is  a  fraction,  in  which  the  denominator  contains  a  fraction,  which 
again  in  its  denominator  contains  a  fraction,  etc.     Thus: 


2+  1 


3+  1 


1+1 

4 


Only  integer  chain  fractions,  that  is,  chain  fractions  in  which 
all  numerators  are  unity,  are  of  interest. 

A  common  fraction  is  converted  into  a  chain  fraction  thusly: 


APPROXIMATION  BY  CHAIN  FRACTION.  208a 

511 


1152       • 

511            1 

1 

1152       1152 
511 

2  +  l^« 
^  511 

1 
"2  +  1 

511 
130 

1 
2+  1 

^+  130 

1 
-2+1 

3+  1 
130 
121 

1 

2  +  1 

3+  1 

^+121 

1 

1 

~  2  +  1         ~         2  +  1 

3+1                    3+1 
1+1                 r+  1 

1                        1 
"2+1         ~         2+1 

3+1                    3+1 

1+1                    1+1 

13+1                 13+1 

1                           2  +  i 
4                                  4 

That  is,  to  convert  a  common  fraction  into  a  chain  fraction, 
the  numerator  is  divided  into  the  denominator,  the  residue 
divided  into  the  divisor,  and  so  on,until  no  residue  remains. 
The  successive  quotients  then  are  the  successive  denominators 
of  the  chain  fraction. 

For  instance: 


2086  ENGINEERING  MATHEMATICS 

511    ^  ^ 


1152 

• 

511/1152  =  2 
1022 

130/511 
390 

=  3 

121/130  =  1 
121 

9/121  =  13 
9 

31 

27 

4/9  =  2 
8 

1/4  = 

4 

hence: 


511 

1 

1152  ■ 

"2  +  1 

3+  1 

1  +  1 


13  4-  1 


2+1 
4 


Inversely,   the   chain  fraction  is  converted  into  a  common 
fraction,  by  rolhng  it  up  from  the  end: 


1   9 

2  +  7  =  7 

4   4 

1      4 

1  ~  9 

2  +  4 

13+1 

121 

1 

9 

2  +  j 

1 

9 

13  +  1 

121 

2  +  1 

APPROXIMATION  BY  CHAIN  FRACTION.         208c 
1  130 


1  + 


13  +  1 121 


1 _  121 

]  +  1 ~  130 

13+  1 


^-l 


3  -{-  1 _  511 

1  +  1 ~  130 

13  +  1 


1 _  130 

3  +  1 ~  511 

1+  1 


13+  1 


^  +  1 


2+  1 ^  1152 

3  +  1 511 

1  +  1 


13  +  1 


2  + 


1 

4 

1 _  ^511^ 

2  +  1 ~  1152 

3  +  1 


1  +  1 


13+  1 


^+i 


The  expression  of  the  numerical  value  by  chain  fraction  gives  a 
series  of  successive  approximations.  Thus  the  successive  ap- 
proximation of  the  chain  fraction : 


208d 


ENGINEERING  MATHEMATICS. 
1  511 


2+  1 


1152 


3+  1 


1  +  1 


13+  1 


2  + 


are: 


difference:  =  %: 

(1)1 

2 

1 

2 

=  .5 

.     +  .0564         =  +12.7% 

(2)1 

3 

7 

^  +  1 

=  .42857  .. 

.     -  .0150        =  -3.4% 

(3)1 
2+  1 

4 
~  9 

=  .44444  .. 

.     +  .00086      =  +.194% 

3  + 


(4)  1 


2+  1 


55 
124 


= .443548 


-  .000028    = 


.0068% 


3+  1 


1  +± 
13 


(5)1 


2+1 


114 
257 


=  . 443580. . .     +  . 000004    =  + . 0009  % 


3+  1 


1  +  1 


13  + 


(6)1 


2+1 


511 
1152 


=  .443576 


3+  1 


1  +  1 


13+1 


2  + 


APPROXIMATION  BY  CHAIN  FRACTION.  208e 

As  seen,  successive  approximations  are  alternately  above  and 
below  the  true  value,  and  the  approach  to  the  true  value  is 
extremely  rapid.  It  is  the  latter  feature  which  makes  the  chain 
fraction  valuable,  as  where  it  can  be  used,  it  gives  very  rapidly 
converging  approximations. 

141 B. — Chain  fraction  representing  irrational  numbers,  as 
TT,  e,  etc.,  may  be  endless.     Thus: 

""  ^  ^  ~^  I- =3.14159265... 

7  +  1 


15+  1 

1  +  1 


288+  1 


1  +  1 


2+  1 


1  +  1 


3  +  1 


1  +  1 


7+  . 


The  first  three  approximations  of  this  chain  fraction  of  tt  are: 

difference :    =  % 
(1)3  +  ^        =3  1/7       =3.142857   .  ..  +  .00127      =  +  .043% 

(2)  3  +  1 15/106  =  3.1415094.  .  .-. 0000832  =-. 0026 % 

'  +  rs 

(3)  3+  1 =3  16/113  =  3.1415929.  .    +.0000003 

7  +  1 =  +  .000009% 

15  +  1 
1 

As  seen,  the  first  approximation,  3  1/7,  is  already  sufficiently 
close  for  most  practical  purposes,  and  the  third  approximation 
of  the  chain  fraction  is  correct  to  the  6th  decimal. 

144. — Frequently  irrational  numbers,  such  as  square  roots, 
can  be  expressed  by  periodic  chain   fractions,   and   the   chain 


208/  ENGINEERING  MATHEMATICS. 

fraction  offers  a  convenient  way  of  expressing  numerical  values 
containing  square  roots,  and  deriving  their  approximations. 

For  instance: 

Resolve  \/6  into  a  chain  fraction. 

As  the  chain  fraction  is  <  1,  \/6  has  to  be  expressed  in  the  form: 

Ve  =  2  +  (Ve  -  2)  (1) 

and  the  latter  term:  (\/6  —  2),  which  is  <1,  expressed  as  chain 
fraction. 

To  rationalize  the  numerator,   we  multiply  numerator  and 
denominator  by  (\/6  +  2): 

,    /-      ^.       (V6-2)(V6  +  2)  2 1_ 

^^^        ^"  V6  +  2  ~V6  +  2"V6  +  2 

2 
thus: 

\/6  =  2 


\/6  +  2 


as is  >  1,  it  is  again  resolved  into: 

\/6  +  2  _  o   ,    \/6-2 
—  2  ^  +  ~2 

thus: 

Ve  =  2  +  1 


2  +  ^^^ 


continuing  in  the  same  manner: 

\/6-2  ^  (V6-2)(\/6  +  2)  ^  2  ^  ^l_^ 

2  2(V6  +  2)  2(\/6  +  2)       Ve -|- 2 

hence: 

Ve  =  2  +  1 


2+  1 


V6  +  2 
and: 

V6  +  2  =  4  +  (V6-2) 
hence: 


APPROXIMATION  BY  CHAIN  FRACTION.         208g 

a/6  =  2  +  1 

2+  1 


4  +  ( V6  -  2) 

and,  as  the  term  (\/q  —  2)  appeared  already  at  (1),  we  are  here 
at  the  end  of  the  recurring  period,  Ihat  is,  the  denominators  now 
repeat: 

Ve  =  2  +  1 

2  +  1 


4+  1 


2+  1 


4  +  1 


2  + 


a  periodic  chain  fraction,  in  which  the  denommators  2  and  4 
alternate. 

In  the  same  manner, 
\/2  =1  +  1  with  the  periodic  denominator  2 

2T1 


2+  1 


2+  . 
\/3  =1  +  1  with  the  periodic  denominators  1  and  2 

r+1 


2+  1 


1+  1 


2+.^ 
with  the  periodic  denominator  4 

4+1 

4+  . 

This  method  of  resolution  of  roots  into  chain  fractions  gives 
a  convenient  way  of  deriving  simple  numerical  approximations 
of  the  roots,  and  hereby  is  very  useful. 

For  instance,  the  third  approximation  of  \/2is  1  ^{2,  with  an 
error  of  .2  per  cent,  that  is,  close  enough  for  most  practical 
purposes.  Thus,  the  diagonal  of  a  square  with  1  foot  as  side, 
is  very  closely  1  foot  5  inches,  etc. 


CHAPTER   VI. 
EMPIRICAL  CURVES. 

A,  General. 

142.  The  results  of  observation  or  tests  usually  are  plotted 

in  a  curve.     Such  curves,  for  instance,  are  given  by  the  core 

loss  of  an  electric  generator,  as  function  of  the  voltage;    or, 

the  current  in  a  circuit,  as  function  of  the  time,  etc.     AMien 

plotting  from  numerical  observations,  the  curves  are  empirical, 

and  the  first  and  most   important    problem  which  has  to  be 

solved  to  make  such  curves  useful  is  to  find  equations  for  the 

same,  that  is,  find  a   function,   y=f{x),  which   represents  the 

curve.     As  long  as  the  equation  of  the  curve  is  not  known  its 

utility  is  very  limited.     While  numerical  values  can  be  taken 

from  the  plotted  curve,  no  general  conclusions  can  be  derived 

from  it,  no  general  investigations  based  on  it  regarding  the 

conditions  of  efficiency,  output,  etc.     An  illustration  hereof  is 

affordetl  by  the  comparison  of  the  electric  and  the  magnetic 

circuit.     In  the  electric  circuit,  the  relation  between  e.m.f.  and 

e 
current  is  given  by  Ohm's  law,  i  =  -  and  calculations  are  uni- 

r 

versally  and  easily  made.  In  the  magnetic  circuit,  however, 
the  term  corresponding  to  the  resistance,  the  reluctance,  is  not 
a  constant,  and  the  relation  between  m.m.f.  and  magnetic  flux 
cannot  be  expressed  by  a  general  law,  but  only  by  an  empirical 
curve,  the  magnetic  characteristic,  and  as  the  result,  calcula- 
tions of  magnetic  circuits  cannot  be  made  as  conveniently  and 
as  general  in  nature  as  calculations  of  electric  circuits. 

If  by  observation  or  test  a  number  of  corresponding  values 
of  the  independent  variable  x  and  the  dependent  variable  y  are 
determined,  the  problem  is  to  find  an  equation,  y=f{x),  which 
represents  these  corresponding  values:  X\,  X2,  X3  . , .  Xn,  and 
yi,  2/2,  2/3  ••  .  ?/n,  approximately,  that  is,  within  the  errors  of 
observation. 

209 


210  ENGINEERING  MATHEMATICS. 

The  mathematical  expression  which  represents  an  empirical 
curve  may  be  a  rational  equation  or  an  empirical  equation. 
It  is  a  rational  equation  if  it  can  be  derived  theoretically  as  a 
conclusion  from  some  general  law  of  nature,  or  as  an  approxima- 
tion thereof,  but  it  is  an  empirical  equation  if  no  theoretical 
reason  can  be  seen  for  the  particular  form  of  the  equation. 
For  instance,  when  representing  the  dying  out  of  an  electrical 
current  in  an  inductive  circuit  by  an  exponential  function  of 
time,  we  have  a  rational  equation:  the  induced  voltage,  and 
therefore,  by  Ohm's  law,  the  current,  varies  proportionally  to  the 
rate  of  change  of  the  current,  that  is,  its  differential  quotient, 
and  as  the  exponential  function  has  the  characteristic  of  being 
proportional  to  its  differential  quotient,  the  exponential  function 
thus  rationally  represents  the  dying  out  of  the  current  in  an 
inductive  circuit.  On  the  other  hand,  the  relation  between  the 
loss  by  magnetic  hysteresis  and  the  magnetic  density :  W=7jB^'^, 
is  an  empirical  equation  since  no  reason  can  be  seen  for  this 
law  of  the  1.0th  power,  except  that  it  agrees  with  the  observa- 
tions. 

A  rational  equation,  as  a  deduction  from  a  general  law  of 
nature,  applies  universally,  within  the  range  of  the  observa- 
tions as  well  as  beyond  it,  while  an  empirical  equation  can  with 
certainty  be  relied  upon  only  within  the  range  of  observation 
from  which  it  is  derived,  and  extrapolation  beyond  this  range 
becomes  increasingly  uncertain.  A  rational  equation  there- 
fore is  far  preferable  to  an  empirical  one.  As  regards  the 
accuracy  of  representing  the  observations,  no  material  difference 
exists  between  a  rational  and  an  empirical  equation.  An 
empirical  equation  frequently  represents  the  observations  with 
great  accuracy,  while  inversely  a  rational  equation  usually 
does  not  rigidly  represent  the  observations,  for  the  reason  that 
in  nature  the  conditions  on  which  the  rational  law  is  based  are 
rarely  perfectly  fulfilled.  For  instance,  the  representation  of  a 
decaying  current  by  an  exponential  function  is  based  on  the 
assumption  that  the  resistance  and  the  inductance  of  the  circuit 
are  constant,  and  capacity  absent,  and  none  of  these  conditions 
can  ever  be  perfectly  satisfied,  and  thus  a  deviation  occurs  from 
the  theoretical  condition,  by  what  is  called  "  secondary  effects." 
143.  To  derive  an  equation,  which  represents  an  empirical 
curve,  careful  consideration  should  first  be  given  to  the  physical 


EMPIRICAL  CURVES.  211 

nature  of  the  phenomenon  which  is  to  be  expressed,  since 
thereby  the  number  of  expressions  which  may  be  tried  on  the 
empirical  curve  is  often  greatly  reduced.  Much  assistance  is 
usually  given  by  considering  the  zero  points  of  the  curve  and 
the  points  at  infinity.  For  instance,  if  the  observations  repre- 
sent the  core  loss  of  a  transformer  or  electric  generator,  the 
curve  must  go  through  the  origin,  that  is,  y  =  0  for  x  =  0,  and 
the  mathematical  expression  of  the  curve  y  =f{x)  can  contain 
no  constant  term.  Furthermore,  in  this  case,  with  increasing  x, 
1/ must  continuously  increase,  so  that  for  x  =  00,  y  =  cc.  Again, 
if  the  observations  represent  the  dying  out  of  a  current  as 
function  of  the  time,  it  is  obvious  that  for  x  =  oo,  ^=0.  In 
representing  the  power  consumed  by  a  motor  when  running 
without  load,  as  function  of  the  voltage,  for  x  =  0,  y  cannot  be 
=  0,  but  must  equal  the  mechanical  friction,  and  an  expression 
like  y  =  Ajif  cannot  represent  the  observations,  but  the  equation 
must  contain  a  constant  term. 

Thus,  first,  from  the  nature  of  the  phenomenon,  which  is 
represented  by  the  empirical  curve,  it  is  determined 

(a)  Whether  the  curve  is  periodic  or  non-periodic. 

(6)  Whether  the  equation  contains  constant  terms,  that  is, 
for  x  =  0,  yy^O,  and  inversely,  or  whether  the  curve  passes 
through  the  origin:  that  is,  ^  =  0  for  x  =  0,  or  w^hether  it  is 
hyperbolic;  that  is,  y=  ao  for  x  =  0,  or  a-=oo  for  ?/==0. 

(c)  What  values  the  expression  reaches  for  oo.  That  is, 
whether  for  x  =  oo,  i/  =  oo,  or  ?/  =  0,  and  inversely. 

(d)  Whether  the  curve  continuously  increases  or  decreases,  or 
reaches  maxima  and  minima. 

(e)  Whether  the  law  of  the  curve  may  change  within  the 
range  of  the  observations,  by  some  phenomenon  appearing  in 
some  observations  which  does  not  occur  in  the  other.  Thus, 
for  instance,  in  observations  in  which  the  magnetic  density 
enters,  as  core  loss,  excitation  curve,  etc.,  frequently  the  curve 
law  changes  with  the  beginning  of  magnetic  saturation,  and  in 
this  case  only  the  data  below  magnetic  saturation  would  be  used 
for  deriving  the  theoretical  equations,  and  the  effect  of  magnetic 
saturation  treated  as  secondary  phenomenon.  Or,  for  instance, 
when  studying  the  excitation  current  of  an  induction  motor, 
that  is,  the  current  consumed  when  running  light,  at  low 
voltage  the  current  may  increase  again  with  decreasing  voltage, 


212  ENGINEERING  MATHEMATICS. 

instead  of  decreasing,  as  result  of  the  friction  load,  when  the 
voltage  is  so  low  that  the  mechanical  friction  constitutes  an 
appreciable  part  of  the  motor  output.  Thus,  empirical  curves 
can  be  represented  by  a  single  equation  only  when  the  physical 
conditions  remain  constant  within  the  range  of  the  observations. 

From  the  shape  of  the  curve  then  frequently,  with  some 
experience,  a  guess  can  be  made  on  the  probable  form  of  the 
equation  which  may  express  it.  In  this  connection,  therefore, 
it  is  of  the  greatest  assistance  to  be  familiar  with  the  shapes  of 
the  more  common  forms  of  curves,  by  plotting  and  studying 
various  forms  of  equations  y=f{jc). 

By  changing  the  scale  in  which  observations  are  plotted 
the  apparent  shape  of  the  curve  may  be  modified,  and  it  is 
therefore  desirable  in  plotting  to  use  such  a  scale  that  the 
average  slope  of  the  curve  is  about  45  deg.  A  much  greater  or 
much  lesser  slope  should  l)e  avoided,  since  it  does  not  show  the 
character  of  the  curve  as  well. 

B.  Non-Periodic  Curves. 

144.  The  most  common  non-periodic  curves  are  the  potential 
series,  the  parabolic  and  hyperbolic  curves,  and  the  exponential 
and  logarithmic  curves. 

The  Potential  Series. 

Theoretically,  any  set  of  observations  can  be  represented 
exactly  by  a  potential  series  of  any  one  of  the  following  forms : 

y  =  aQ  +  aiX  +  a2.r^+a3X^-\^ .  .  .  ■.      .     .     .     .     (1) 

y=-aix+a2X-+o^x^  +  .  .  .  ; (2) 

a  1     rt>     0.3 
y  =  «o+-+-,+j3  +  ...  : (3) 

y—+7^+r^+ w 

if  a  sufficiently  large  number  of  terms  are  chosen. 

For  instance,  if  n  corresponding  numerical  values  of  x  and  y 
are  given,  Xi,  yi]   X2,  y^;    •  •  •  ^n,  y.r,  they  can  be  represented 


EMPIRICAL  CURVES. 


213 


by  the  series  (1),  when  choosing  as  many  terms  as  required  to 
f^ive  n  constants  a: 


y  =  Go  +  aix -\- a2Jc^ -h .  .  .+a„_in"~^ 


(5) 


By  substituting  the  corresponding  values  Xi,  yi)  X2,  y2,  ■  ■  • 
into  equation  (5),  there  are  obtained  n  equations,  which  de- 
termine the  n  constants  ao,  ai,  a2,  .  .  .  an_i. 

Usually,  however,  such  representation  is  irrational,  and 
therefore  meaningless  and  useless. 

Table  I. 


e 
100  ""^ 

Pv  =  i/ 

-0.5 

+  2x 

+  2.5x2 

-1.5x3 

+  1.5x< 

-2xS 

+  x> 

0.4 
0  6 
0.8 

0.63 
1.36 

2.18 

-0.5 
-0.5 
-0.5 

+  0.8 
+  1.2 
+  1.6 

+  0.4 
+  0.9 
+  1.6 

-0.10 
-0.32 
-0.77 

+0.04 
+0.19 
+  0.61 

-  0.02 

-  0.16 

-  0.65 

0 
+   0.05 
+   0.26 

1.0 
1.2 
1.4 

3.00 
3.93 
6.22 

-0.5 
-0.5 
-0.5 

+  2.0 

+  2.4 
+  2.8 

+  2.5 
+  3.6 
+  4.9 

-1.50 
-2.59 
-4.12 

+  1.50 
+  3.11 

+  5.76 

-  2.00 

-  4.98 
-10.76 

+   1.00 
+  2.89 
+   6.13 

1.6 

S.50 

-0.5 

+3.2 

+  6.4 

-6.14 

+  9.83 

-20.97 

+  16.78 

Let,  for  instance,  the  first  column  of  Table  I  represent  the 
voltage,  Ycici^''^'  ^^  hundreds  of  volts,  and  the  second  column 

the  core  loss,  pi=^y,  in  kilowatts,  of  an  125-volt  100-h.p.  direct- 
current  motor.  Since  seven  sets  of  observations  are  given, 
they  can  be  represented  by  a  potential  series  with  seven  con- 
stants, thus, 

y  =  ao-\-aiX+a2X^+.  .  .+aQX^,     ....     (6) 

and  by  substituting  the  observations  in  (G),  and  calculating  the 
constants  a  from  the  seven  equations  derived  in  this  manner, 
there  is  obtained  as  empirical  expression  of  the  core  loss  of 
the  motor  the  equation, 


?/= -0.5 +2j+2.5.r2- 1.5x3 +  1.5j4-2.r5+j« 


(7 


This  expression  (7),  however,  while  exactly  representing 
the  seven  observations,  has  no  physical  meaning,  as  easily 
seen  by  plotting  the  individual  terms.     In  Fig.  60,  y  appears 


214 


ENGINEERING  MATHEMATICS. 


as  the  resultant  of  a  number  of  large  positive  and  negative 
terms.  Furthermore,  if  one  of  the  observations  is  omitted, 
and  the  potential  series  calculated  from  the  remaining  six 
values,  a  series  reaching  up  to  x^  would  be  the  result,  thus, 

y  =  aQ-\-a\x-\-a2X^-V.  .  .-\-ar,7^,     ....     (8) 


16 

12 

/ 

f 

1 

8 

i 

r 

_^ 

4 

<^ 

^ 

^ 

*^ 

y^ 

^ 

^ 

^ 

♦fa 

~ 

0 



= 

B= 

:=S 

^ 

^ 

^ 

— 

^ 

' — , 

-0.5 

-4 

N 

s. 

■^ 

^*^. 

\ 

s. 

"V 

-8 

\ 

Vv> 

^ 

f 

-12 

\ 

\ 

\ 

.16 

\ 

1 

\ 

.20 

x  = 

\ 

0 

2 

0 

.4 

0 

6 

0 

8 

1 

0 

1 

2 

1 

A 

I 

8 

Fig.  60.    Terms  of  Empirical  Expression  of  Excitation  Power. 

but  the  constants  a  in  (8)  would  have  entii'ely  different  numer- 
ical values  from  those  in  (7),  thus  showing  that  the  equation 
(7)  has  no  rational  meaning. 

145.  The  potential  series  (1)  to  (4)  thus  can  be  used  to 
represent  an  empirical  curve  only  under  the  following  condi- 
tions : 

1.  If  the  successive  coefficients  ao,  a\,  ao,  ...  decrease  in 
value  so  rapidly  that  within  the  range  of  observation  the 
higher  terms  become  rapidly  smaller  and  appear  as  mere 
secondary  terms. 


EMPIRICAL  CURVES. 


215 


2.  If  the  successive  coefficients  a  follow  a  definite  law, 
indicating  a  convergent  series  which  represents  some  other 
function,  as  an  exponential,  trigonometric,  etc. 

3.  If  all  the  coefficients,  a,  are  very  small,  with  the  exception 
of  a  few  of  them,  and  only  the  latter  ones  thus  need  to  be  con- 
sidered. 

Table  II. 


X 

V 

v' 

V\ 

0.4 

0.89 

0.88 

0.01 

0.6 

1.35 

1.34 

0.01 

0.8 

1.96 

1.94 

0.02 

1.0 

2.72 

2.70 

0.02 

1.2 

3.62 

2.59 

0.03 

1.4 

4.63 

4.59 

0.04 

1.6 

5.76 

5.65 

0.11 

For  instance,  let  the  numbers  in  column  1  of  Table  II 
represent  the  speed  or  of  a  fan  motor,  as  fraction  of  the  rated 
speed,  and  those  in  column  2  represent  the  torque  y,  that  Is, 
the  turning  moment  of  the  motor.  These  values  can  be 
represented  by  the  equation, 

i/ =  0.5 +0.022:+2.5j:2-0.3x-^ +  0.015x^-0.02x5 +o.01.r6.     (9) 

In  this  case,  only  the  constant  term  and  the  terms  with 
x^  and  x^  have  appreciable  values,  and  the  remaining  terms 
probably  are  merely  the  result  of  errors  of  observations,  that  is, 
the  approximate  equation  is  of  the  form. 


y  =  ao+a2X^+a3X^ 

Using  the  values  of  the  coefficients  from  (9),  gives 
1/  =  0.5 +  2.5x2 -0.3x3.      ^     ^     ^ 


(10) 


(11) 


The  numerical  values  calculated  from  (11)  are  given  in  column 
3  of  Table  II  as  y',  and  the  difference  between  them  and  the 
observations  of  column  2  are  given  in  column  4,  as  yi. 


216  ENGINEERING  MATHEMATICS. 

The  values  of  column  4  can  now  be  represented  by  the  same 
form  of  equation,  namely, 

t/l=6o+&2X2+63^^ (12) 

in  which  the  constants  &o,  &2,  &3  are  calculated  by  the  method 
of  least  squares,  as  described  in  paragraph  120  of  Chapter  IV, 
and  give 

?/i  =0.031 -0.093.r2  +  0.07r)r'' (13) 

Equation  (13)  added  to  (11)  gives  the  final  approximate 
equation  of  the  torque,  as, 

1/0  =  0.531 +2.407x2- 0.224  j^^ (14) 

The  equation  (14)  probably  is  the  approximation  of  a 
rational  equation,  since  the  first  term,  0.531,  represents  the 
bearing  friction;  the  second  term,  2.407x2  (which  is  the  largest), 
represents  the  work  done  by  the  fan  in  moving  the  air,  a 
resistance  proportional  to  the  square  of  the  speed,  and  the 
third  term  approximates  the  decrease  of  the  air  resistance  due 
to  the  churning  motion  of  the  air  created  by  the  fan. 

In  general,  the  potential  series  is  of  limited  usefulness;  it 
rarely  has  a  rational  meaning  and  is  mainly  used,  where  the 
curve  approximately  follows  a  simple  law,  as  a  straight  line, 
to  represent  by  small  terms  the  deviation  from  this  simple  law, 
that  is,  the  secondary  effects,  etc.  Its  use,  thus,  is  often 
temporary,  giving  an  empirical  apj)roximation  pending  the 
derivation  of  a  more  rational  law. 

The  Parabolic  and  the  Hyperbolic  Curves. 

146.  One  of  the  most  useful  classes  of  curves  in  engineering 
are  those  represented  by  the  equation, 

2/  =  a-r"; (15) 

or,  the  more  general  equation, 

y-b  =  a{x-cy (16) 

Equation  (16)  differs  from  (15)  only  by  the  constant  terms  b 
and  c;   that  is,  it  gives  a  different  location  to  the  coordinate 


EMPIRICAL  CURVES. 


217 


center,  but  the  curve  shape  is  the  same,  so  that  in  discussing 
the  general  shapes,  only  equation  (15)  need  be  considered. 

If  n  is  positive,  the  curves  y  =  ax'^  are  parabolic  curves, 
passing  through  the  origin  and  increasing  with  increasing  x. 
U  n>l,  y  increases  with  increasing  rapidity,  U  n<l,y  increases 
with  decreasing  rapidity. 

If  the  exponent  is  negative,  the  curves  y=ax~^= —  are 

hyperbolic  curves,  starting  from  2/ =  00  for  x=0,  and  decreasing 
to  t/=0  for  x=  00. 

n-=^\  gives  the  straight  line  through  the  origin,  n=0  and 
n=oo  give,  respectively,  straight  horizontal  and  vertical  lines. 

Figs.  61  to  71  give  various  curve  shapes,  corresponding  to 
different  values  of  n. 


Parabolic  Curves. 
Fig.  61.  n  =  2 
Fig.  62.  n  =  4 
Fig.  63.  n  =  8 
Fig.  64.  n  =  i 
Fig.  65.  n  =  \ 
Fig.  66.     n  =  i 

Hyperbolic  Curves 
Fiji.  67. 


y  =  x^ 
y  =  x^ 


the  common  parabola, 
the  biquadratic  parabola. 


y=Vx;  again  the  common  parabola. 
y=  '^;  the  biquadratic  parabola. 
y  =  \/x. 


1 


—  1;     y  =  -',  the  equilateral  hyperbola. 


Fig.  68. 
Fig.  69. 
Fig.  70. 
Fig.  71. 


n=-2: 


y=-r 


n=-4;    y 


1  1 


-4;  y- 


Vx' 

1 

'Ix' 


218 


ENGINEERING  MATHEMATICS. 


"~~ 

" — 

— 



— 

■— 



^ 

\ 

s. 

\ 

\ 

' 

< 

5 

o 

7 
* 

CO 

^ 

H 

0 

■J 

o 

CO 
O 

o 
o 

c5 

O 

— 

— 

■~~- 

^ 

^ 

^^ 

'  ■*-, 

"^ 

.^ 

X 

s 

V 

\, 

s 

\ 

\ 

\ 

c 

i» 

° 

3 

4 

« 

3 
H 

■^ 

c 

e 

3 

C 

0 

5 

3 

5 

c 

3 

c 

1 

i 

^^ 

^ 

■'^^ 

^ 

■\ 

^ 

^ 

^ 

'^ 

^ 

\ 

N 

N 

\ 

S 

\ 

s 

s 

\ 

\ 

y 

\ 

\ 

< 
< 

> 

: 

) 
4 

c 

3 
4 

4 

« 

) 

< 

5 

4 

J 

< 

3 
3 

^ 

5 

< 

3 

P^ 


3 

CO 

CJ 

o 

y 

O 

^ 

c« 

o 

<-, 

at 

PL, 

-*< 

o 

i-H 

CO 

o 

o 

Pm 

EMPIRICAL  CURVES. 


219 


1 

^ 

^ 

*-- 

^ 

■l.Z 

^ 

-^ 

^ 

^ 

1.0 

^ 

^ 

^ 

^ 

^ 

y 

/ 

y 

/ 

/ 

/ 

/ 

OHi 

/ 

y 

/ 

0 

2 

0 

i 

0 

6 

0 

8 

1 

0 

1 

2 

1 

1 

I 

6 

I 

8 

29 

Fig.  64.     Parabolic  Curve.     y  =  \^lc. 


►1-0- 

. . 

■ 

— ' 

1,0- 

-^ 

i^^** 



"^ 

0-8- 

^ 

^ 

^ 

^ 

0-6- 

/ 

p^ 

/ 

/ 

f. 

0:?- 

0 

2 

0 

4 

0 

6 

0 

8 

1 

0 

1 

2 

1 

4 

1 

6 

1 

8 

2 

0 

Fig.  65.     Parabolic  Curve.     y=  yjx. 


220 


ENGINEERING  MATHEMATICS. 


__ 

■ 



"^ 

^ 



^ 

^^^ 

/ 

/ 

0 

2 

0 

4 

0 

6 

0 

8 

1 

0 

1^ 

1 

4 

1 

6 

1 

8 

2 

0 

Fig.  66.     Parabolic  Curve.     y  =  \^x 


3tS*- 

\ 

\ 

-2-A- 

\ 

\ 

> 

V 

\ 

\ 

s. 

\ 

N 

s 

^ 

^ 

^_____ 

0 

^ 

0 

8 

1 

2 

i 

1 

S 

2 

" 

2 

4 

2 

8 

3 

2 

3 

6 

4,0 

Fig.  67.     Hyperbolic  Curve  (Equilateral  Hyperbola).     y  =  —  - 

X 


EMPIRICAL  CURVES. 


221 


ifcSS- 

2.4- 

\ 

\ 

\ 

y 

\ 

\ 

V 

s 

s. 

\ 

^ 

— 

0.4  0.8  1.2  1.6         .2.0  2.4  2.8  3.2  3.6  4.0  4.4 


F 

IG. 

68. 

Hyperbolic  Curve. 

y 

x' 

2.ft- 

1 

, 

1  fi 

fc2- 

\ 

\ 

\ 

\ 

^ 

\, 

\ 

s 

"^ 

0.4  0.8  1.2  1.6  2.0  2.4  2.8 


3.0  4.0  4.1 


Fig.  69.     Hyperbolic  Curve.     2/ =  -4- 


222 


ENGINEERING  MATH  EM  A  TICS. 


T 

1 

\ 

\ 

\, 

s 

\ 

s 

X 

■^ 

-«^ 



. 



~ 

' 





■ 

0 

4 

0 

8 

1 

2 

1 

6 

8 

0 

2 

4 

2 

8 

3 

2 

3 

6 

4 

0 

4 

4 

r 

Fig.  70.     Hyperbolic  Curve.     j/=— =. 

274 

T 

A 

l.C\ 

\ 

\ 

s. 

V 

"^ 

^~~- 



_      _ 

~ 

0 

i 

0 

8 

1 

2 

1 

6 

2 

0 

2 

4 

2 

8 

3 

2 

3 

6 

4 

0 

4 

4 

Fig.  71.     Hyperbolic  Curve.     y  =  — —. 


EMPIRICAL  CURVES.  223 

In  Fig.  72,  sixteen  different  parabolic  and  hyperbolic  curves 
are  drawn  together  on  the  same  sheet,  for  the  following  values : 
n  =  l,  2,  4,  8,  o);   i  i  i,  0;    -1,  -2,  -4,  -8;    -h,  -i,  -^ 

147.  Parabolic  and  hyperbolic  curves  may  easily  be  recog- 
nized by  the  fact  that  if  x  is  changed  by  a  constant  factor,  y  also 
changes  by  a  constant  factor. 

Thus,  in  the  curve  y  =  x-,  doubhng  the  x  increases  the  y 
fourfold;  in  the  curve  y  =  x^-^^,  doubling  the  x  increases  the  y 
threefold,  etc.;  that  is,  if  in  a  curve, 

y=f(x), 

-^^  =  constant,  for  constant  g,      .     .     .     (17) 

the  curve  is  a  parabolic  or  hyperbolic  curve,  y  =  ax"",  and 

fiqx)     aiqx)"^ 

f(x)^'~^^^'^^ ^^^^ 

If  q  is  nearly  1,  that  is,  the  x  is  changed  onjy  by  a  small 
value,  substituting  g  =  l+s,  where  s  is  a  small  quantity,  from 
equation  (18), 

fix+sx)     ^^      , 

hence, 

fix+sx)-f{x) 

fix)  ""'' ^^^^ 

that  is,  changing  x  by  a  small  percentage  s,  y  changes  oy  a  pro- 
portional small  percentage  ns. 

Thus,  parabolic  and  hyperbolic  curves  can  be  recognized  by 
a  small  percentage  change  of  x,  giving  a  proportional  small 
percentage  change  of  y,  and  the  proportionality  factor  is  the 
exponent  n;  or,  they  can  be  recognized  by  doubling  x  and 
seeing  whether  y  hereby  changes  by  a  constant  factor. 

As  illustration  are  shown  in  Fig.  73  the  parabolic  curves, 
which,  for  a  doubling  of  x,  increase  y:  2,  3,  4,  5,  6,  and  8  fold. 

Unfortunately,  this  convenient  way  of  recognizing  parabolic 
and  hyperbolic  curves  appUes  only  if  the  curve  passes  through 
the  origin,  that  is,  has  no  constant  term.  If  constant  terms 
exist,  as  in  equation  (16),  not  x  and  y,  but  (x—c)  and  (y—b) 
follow  the  law  of  proportionate  increases,  and  the  recognition 


224 


ENGINEERING  MATHEMATICS. 


becomes  more  difficult;    that  is,  various  values  of  c  and  of  h 
are  to  be  tried  to  find  one  whicli  o;ives  the  proportionaht}'. 


0^  0,4  0.6  0.8  1.0  1.2  1.4 

Fig.  72.     Parabolic  and  Hyperbolic  Curves.     y  =  xn, 
148.  Taking  the  logarithm  of  equation  (15)  gives 

log  2/  =  log  a  +  /i  log  x; (20) 


EMPIRICAL  CURVES.  225 

that  is,  a  straight  line;  hence,  a  parabolic  or  hyperbolic  curve  can 
be  recognized  by  plotting  the  logarithm  of  y  against  the  loga- 
rithm of  X.  If  this  gives  a  straight  line,  the  curve  is  parabolic 
or  hyperbolic,  and  the  slope  of  the  logarithmic  curve,  tan  0  =  n, 
is  the  exponent. 


J 

1 1 0 

/ 

02  r 
"  1 

^ 
t'l 

t 

1 

1      fs 

// 

7 

/ 

// 

/ 

/ 

// 

/ 

fco 

/ 

/ 

/ 

\ 

^ 

1 

y 

/ 

1 

m 

/ 

/ 

It 

V 

/ 

f/ 

'/ 

r 

IfU 

A 

fl 

/ 

S. 

i 

/ 

/ 

w 

/ 

/ 

y^ 

' 

/ 

/ 

Va 

' 

/ 

0 

^// 

/ 

/ 

/ 

// 

^ 

y 

05 

/ 

y 

->, 

'4 

y 

^ 

^ 

^■^ 

^ 

y 

0.2  0.4  0.6  0.8  1.0  1.2  1.4  1.6 

Fig.  73.     Parabolic  Curves,     y^xn. 

This  again  applies  only  if  the  curve  contain  no  constant 
term.  If  constant  terms  exist,  the  logarithmic  line  is  curved. 
Therefore,  by  trying  different  constants  c  and  6,  the  curvature 
of  the  logarithmic  line  changes,  and  by  interpolation  such 
constants  can  be  found,  which  make  the  logarithmic  line  straight, 
and  in  this  way,  the  constants  c  and  h  may  be  evaluated.  If 
only  one  constant  exist,  that  is,  only  6  or  only  c,  the  process  is 
relatively  simple,  but  it  becomes  rather  complicated  with  both 


226  EXGINEERING  MATHEMATICS. 

constants.    This  fact  makes  it  all  the  more  desirable  to  get 
from  the  physical  nature  of  the  problem   some  idea  on  the 
existence  and  the  value  of  the  constant  terms. 
Differentiating  equation  (20)  gives : 

du      dx 
y       ^ 

that  is,  in  a  parabolic   or  hyperbohc   curve,   the  percentual 
change,  or  variation  of  y,  is  n  times  the  percentual  change, 
or  variation  of  x,  if  n  is  the  exponent. 
Herefrom  follows: 

dy 

y 

dx 

X 

that  is,  in  a  parabolic  or  hyperbolic  curve,  the  ratio  of  variation, 
dy 

y 

m  =  -—-,  is  a  constant,  and  equals  the  exponent  n. 
dx 

X 

Or,  inversely: 

If  in  an  empirical  curv^e  the  ratio  of  variation  is  constant 
the  curve  is — within  the  range,  in  which  the  ratio  of  variation 
is  constant — a  parabohc  or  hyperbohc  curve,  which  has  as 
exponent  the  ratio  of  variation. 

In  the  range,  however,  in  which  the  ratio  of  variation  is 
not  constant,  it  is  not  the  exponent,  and  while  the  empirical 
curve  might  be  expressed  as  a  parabohc  or  hyperbolic  curve 
with  changing  exponent  (or  changing  coefficient),  in  this  case 
the  exponent  may  be  very  different  from  the  ratio  of  varia- 
tion, and  the  change  of  exponent  frequently  is  very  much 
smaller  than  the  change  of  the  ratio  of  variation. 

This  ratio  of  variation  and  exponent  of  the  parabohc  or 
hyperbohc  approximation  of  an  empirical  curve  must  not  be 
mistaken  for  each  other,  as  has  occasionally  been  done  in 
reducing  hysteresis  curves,  or  radiation  curv^es.    They  coincide 


EMPIRICAL  CURVES.  227 

only  in  that  range,  in  which  exponent  n  and  coefficient  a  of 
the  equation  y^ax"  are  perfectly  constant.  If  this  is  not 
the  case,  then  equation  (20)  differentiated  gives: 

dy    da    .         ,       n 

—  = 1- log  X  dn  +  ~dx, 

y       a  X 

and  the  ratio  of  variation  thus  is: 

dy 

y  X  da        .         dn 

m  =  —-  =  n-\ \-x\os,x  — : 

dx  ax  X 

X 

that  is,  the  ratio  of  variation  m  differs  from  the  exponent  n. 

Exponential  and  Logarithmic  Curves. 

149.  A  function,  which  is  very  frequently  met  in  electrical 
engineering,  and  in  engineering  and  physics  in  general,  is  the 
exponential  function, 

7/  =  a£'^^; (21) 

which  may  be  written  in  the  more  general  form, 

?y-?;  =  a£"^-^-'=^ (22) 

Usually,  it  appears  with  negative  exponent,  that  is,  in  the 
form, 

y  =  a£-"- (23) 

Fig.  74  shows  the  curve  given  hy  the  exponential  function 
(23)  for  a  =  l;  n^-l;  that  is, 

y=^~% (24) 

as  seen,  with  increasing  positive  x,  y  decreases  to  0  at  a- =  +  00, 
and  with  increasing  negative  x,  y  mcreases  to  00  at  .r=  —  00, 


228 


ENGINEERING  MATHEMATICS. 


The  oiirvo,  7/=£"^^,  has  the  same  shape,  except  that  the 
positive  and  the  negative  side  (right  and  left)  are  interchanged. 

Inverted  these  equations  (21)  to  (24)  may  also  be  written 
thus, 

y. 


nx=loa: 


a' 


n(x-c)  =  log —^; 


V . 


nx  =  —  log  - ; 
^  a 

x^-hgij] 

that  is,  as  logarithmic  curves. 


-2.0        -1.6        -1.2        -0.8        -0.4  0  0.4  0.8  1.2 

Fig.  74.     Exponential  Function,     y^e-x. 


(25) 


\1 

\ 

plT4 

.> 

\ 

. 

\ 

\ 

1.2 

\ 

\ 

\ 

\ 

\ 

\ 

\, 

\ 

Oro 

\ 

\ 

\ 

\ 

\ 

s. 

\ 

s 

\ 

s. 

\ 

\, 

\ 

X 

s 

N 

-I-H 

X 

^ 

^^ 

^ 

■-> 

— 

— 

. 

2.0 


150.  The  characteristic  of  the  exponential  function  (21)  is, 
that  an  increase  of  x  by  a  constant  term  increases  (or,  in  (23) 
and  (24),  decreases)  y  by  a  constant  factor. 

Thus,  if  an  empirical  curve,  y=f{x),  has  such  characteristic 
that 

fix+q) 


fix) 


^constant,  for  constant  q,   . 


(26) 


EMPIRICAL  CURVES. 


229 


the  curve  is  an  exponential  function,  y  =  ae'^^,  and  the  following 
equation  may  be  written : 


f(x  +  q)     a£^(^+g) 
fix)    ~    a£"-=~~ 


gTiq 


(27) 


Hereby  the  exponential  function  can  easily  be  recognized, 
and  distinguished  from  the  parabolic  curve;  in  the  former  a 
constant  terrn,  in  the  latter  a  constant  factor  of  x  causes  a 
change  of  y  by  a  constant  factor. 

As  result  hereof,  the  exponential  curve  with  negative 
exponent  vanishes,  that  is,  becomes  negligibly  small,  with  far 
greater  rapidity  than  the  hyperbolic  curve,  and  the  exponential 


V 

-ItO 

N 

\> 

\ 

C' 

C 

2.4 

\\ 

^. 

0:6 

(  ac+i.S 

5)=^^ 

^ 

V 

^ 

N 

>^, 

^ 

:::::^ 

' 

.4 

2 

LX-i 

-1  55 

£- 

X^ 

^^ 



'      ■ 

— 

~~^" 

0.4  0.8  1.2  1.6  2.0  2.4  2.8  3.2  3.6 

Fig.  75.     Hyperbolic  and  Exponential  Curves  Comparison. 


4.0 


function   with   positive   exponent   reaches   practically   infinite 

values   far  more   rapidly  than  the   parabolic   curve.     This   is 

illustrated    in    Fig.    75,    in    which    are    shown    superimposed 

the  exponential    curve,    y=s~^,    and    the    hyperbolic    curve, 

2.4  ..     . 

y=  .  ,    which   coincides   with   the   exponential   curve 

\X  "r  i.OOJ 

at  x  =  0  and  at  a:  =  l. 

Taking  the  logarithm  of  equation  (21)  gives  logy  — log  a + 
nx  log  £,  that  is,  log  2/  is  a  linear  function  of  x,  and  plotting 
log  y  against  x  gives  a  straight  line.     This  is  characteristic  of 


230 


ENGINEERING  MATHEMATICS. 


the  exponential  functions,  and  a  convenient  method  of  recog- 
nizing them. 

However,  both  of  these  characteristics  apply  only  if  x  and  y 
contain  no  constant  terms.  With  a  single  exponential  function, 
only  the  constant  term  of  y  needs  consideration,  as  the  constant 
term  of  x  may  be  eliminated.  Equation  (22)  may  be  written 
thus: 


—  as' 


^^--nc^nx 


=  A£^, 


(28) 


where  A=ae~^  is  a  constant. 

An  exponential  function  which  contaias  a  constant  term  h 
would  not  give  a  straight  line  when  plotting  log  y  against  x, 


\ 

\ 

\ 

(1)  t/=£-«+o.5S"2a; 

(2)  y=£-oc+o,2£-2x 

U)  2/=£"»-o.2£-2a; 

(5)  y=£-«-o.5£~2» 

(6)  y  =  e-X_Q,^£-2X 

(7)  y  =  £-X-£-'ix 

(8)  y=e-x-i_5£-2x 

y 

1" 

\ 

A 

-hO 

Val 

.\ 

N 

\ 

\ 

-0.-8 

\(4 

i\ 

s\ 

\^ 

\\ 

\ 

-0;& 

S^ 

kN 

k^ 

~" 

■^ 

N 

^ 

s 

(6). 

::^ 

^ 

X 

^ 

^- 

===^^^ 

bv 

-0.2 

^ 

/ 

fjr 

-- 

"^^ 

cg^ 

s== 

/ 

/ 

^e 

^^== 

^=- 

/ 

I 

0 

8 

1 

2 

1 

G 

2 

0 

2 

4 

2 

8 

m 

-0.2 

/ 

/ 

-0.4 

/ 

Fig.  76.     Exponential  Functions. 


EMPIRICAL  CURVES. 


231 


but  would  give  a  curve.  In  this  case  then  log  (y—h)  would  be 
plotted  against  x  for  various  vahies  of  b,  and  by  interpolation 
that  value  of  b  found  which  makes  the  logarithmic  curve  a 
straight  line. 

151.  While  the  exponential  function,  when  appearing  singly, 
is  easily   recognized,   this   becomes   more   difficult   with  com- 


1 
1 

(1)  y=r^+o.6e-io^ 

(2)  y=  £~^ 

(3)  y=£-SC-o,i£-^OX 

(4)  2/=  £"^-0.5  £-10'*' 

(5)  7/=e^a;_£_ioa; 

(6)  2/=£-a'-U£-io« 

\" 

) 

\ 

\A 

(3N 

"A 

k 

r^ 

V 

s 

N 

I 

) 

\ 

k 

\ 

\ 

\ 

^ 

•^ 

. 

0 

4 

0 

8 

1 

2 

1 

6 

2 

0 

2 

4 

2 

8 

LI 


1.2 


1.0 


0.8 


0.6 


0.4 


0.2 


■0.2 


-0.4 


Fig.  77.    Exponential  Functions. 

binations  of  two  exponential  functions  of  different  coefficients 
in  the  exponent,  thus. 


y  =  ais   '''^±a2£   '^, 


(29) 


since  for  the  various  values  of  a\,  02,  c\,  ci,  quite  a  number  of 
various  forms  of  the  function  appear. 

As  such  a  combination  of  two  exponential  functions  fre- 
quently appears  in  engineering,  some  of  the  characteristic  forms 
are  plotted  in  Figs.  76  to  78. 


232  ENGINEERING  MATHEMATICS. 

Fig.  76  gives  the  following  combinations  of  £~^  and  £-2t. 

(2)  ^=£--^+0.2£-2^; 

(3)  ?/=£-^; 

(4)  i/=£-^-0.2£-2^; 

(5)  7/=£---0.5£-2- 

(6)  ?/=£-^-0.S£-2^; 

(7)  ^=,-x_,-2.. 

(8)  i/=£-- 1.5.-2.. 


/ 

r 

/ 

/ 

1 

coshx  =  jie+^+£-«^f 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

t  y 

/ 

/ 

.""^ 

y 

/ 

/ 

^ 

y 

/ 

X 

A 

^ 

,  / 

/ 

.y 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

0 

2 

0 

4 

0 

6 

0 

8 

1 

0 

1 

2 

1 

4 

Fig.  78.     Hyperbolic  Functions. 


EMPIRICAL  CURVES.  233 

Fig.  77  gives  the  following  combination  of  s"^  and  s"^^^: 

(1)  ^=£-^+0.5£-io^; 

(2)  y=e--, 

(3)  ?/=£-^-0.1s-io^; 

(4)  7/=£-^-0.5£-^o^; 

(5)  7/=c-x_.-10r. 

(6)  y=£-^_i.5£-iOx, 

Fig.  78  gives  the  hyperbolic  functions  as  combinations  of 
£"*"'  and  £~^  thus, 

^  =  cosh  x=A(£+-^  +  £-^); 
2/=sinh  j  =  ^(£  +  ^— £-^), 

C.  Evaluation  of  Empirical  Curves. 

152.  In  attempting  to  solve  the  problem  of  finding  a  mathe- 
matical equation,  y=f{x),  for  a  series  of  observations  or  tests, 
the  corresponding  values  of  x  and  y  are  first  tabulated  and 
plotted  as  a  curve. 

From  the  nature  of  the  physical  problem,  which  is  repre- 
sented by  the  numerical  values,  there  are  derived  as  many 
data  as  possible  concerning  the  nature  of  the  curve  and  of  the 
function  which  represents  it,  especially  at  the  zero  values  and 
the  values  at  infinity.  Frequently  hereby  the  existence  or 
absence  of  constant  terms  in  the  equation  is  indicated. 

The  log  X  and  log  y  are  tabulated  and  curves  plotted  between 
X,  y,  log  X,  log  y,  and  seen,  whether  some  of  these  curves  is  a 
straight  line  and  thereby  indicates  the  exponential  function,  or 
the  parabolic  or  hyperbolic  function. 

If  cross-section  paper  is  available,  having  both  coordinates 
divided  in  logarithmic  scale,  and  also  cross-section  paper  having 
one  coordinate  divided  in  logarithmic,  the  other  in  common 
scale,  X  and  y  can  be  directly  plotted  on  these  two  forms  of 
logarithmic  cross-section  paper.  Usually  not  much  is  saved 
thereby,  as  for  the  numerical  calculation  of  the  constants  the 
logarithms  still  have  to  be  tabulated. 


234  ENGINEERING  MATHEMATICS. 

If  neither  of  the  four  curves:  x,  y;  x,  log  y;  log  x,  y;  log  x, 
log  ?/  is  a  straight  line,  and  from  the  physical  condition  the 
absence  of  a  constant  term  is  assured,  the  function  is  neither 
an  exponential  nor  a  parabolic  or  hyperbolic.  If  a  constant 
term  is  probable  or  possible,  curves  are  plotted  between  .t, 
y—b,  logo:,  log  (^—6)  for  various  values  of  b,  and  if  hereby 
one  of  the  curves  straightens  out,  then,  by  interpolation, 
that  value  of  b  is  found,  which  makes  one  of  the  curves  a  straight 
line,  and  thereby  gives  the  curve  law.  A  convenient  way  of 
doing  this  is:  if  the  curve  with  log  y  (cm-ve  0)  is  curved  by  angle 
ao  (cvq  being  for  instance  the  angle  between  the  tangents  at  the 
two  end  points  of  the  curve,  or  the  difference  of  the  slopes  at  the 
two  end  points),  use  a  value  b^,  and  plot  the  ciu^ve  with  log 
iy~bi)  (curve  1),  and  observe  its  cm'vatm-e  a^.  Then  inter- 
polate a  value  62,  between  b^  and  0,  in  proportion  to  the  curva- 
tiu'es  «!  and  ao,  and  plot  curve  with  log  (y  —  b^)  (curve  2),  and 
again  interpolate  a  value  63  between  62  and  either  b^  or  0,  which- 
ever curve  is  nearer  in  slope  to  ciu^ve  2,  continue  until  either  the 
curve  with  log  {y  —  b)  becomes  a  straight  line,  or  an  S  curve  and 
in  this  latter  case  shows  that  the  empuical  curve  cannot  be 
represented  in  this  manner. 

In  this  work,  logarithmic  paper  is  very  useful,  as  it  permits 
plotting  the  curves  without  first  looking  up  the  logarithms,  the 
latter  being  done  only  when  the  last  approximation  of  b  is 
found.  In  the  same  manner,  if  a  constant  term  is  suspected  in 
the  X,  the  value  (x—c)  is  used  and  ciu*ves  plotted  for  various 
values  of  c.  Frequently  the  existence  and  the  character  of  a 
constant  term  is  indicated  by  the  shape  of  the  ciu-ve;  for 
instance,  if  one  of  the  curves  plotted  between  x,  y,  log  x,  log  y 
approaches  straightness  for  high,  or  for  low  values  of  the  ab- 
scissas, but  curves  considerably  at  the  other  end,  a  constant 
term  may  be  suspected,  which  becomes  less  appreciable  at  one 
end  of  the  range.  For  instance,  the  effect  of  the  constant  c  in 
{x—c)  decreases  with  increase  of  x. 

Sometimes  one  of  the  curves  may  be  a  straight  line  at  one 
end,  but  curve  at  the  other  end.  This  may  indicate  the  presence 
of  a  term,  which  vanishes  for  a  part  of  the  observations.  In 
this  case  only  the  observations  of  the  range  which  gives  a 
straight  line  are  used  for  deriving  the  curve  law,  the  curve 
calculated  therefrom,  and  then  the  difference  between  the 
calculated  curve  and  the  observations  further  investigated. 


EMPIRICAL  CURVES.  235 

Such  a  deviation  of  the  curve  from  a  straight  line  may  also 
indicate  a  change  of  the  curve  law,  by  the  appearance  of 
secondary  phenomena,  as  magnetic  saturation,  and  in  this  case, 
an  equation  may  exist  only  for  that  part  of  the  curve  where  the 
secondary  phenomena  are  not  yet  appreciable.  The  same 
equation  may  then  be  applied  to  the  remaining  part  of  the  curve, 
by  assuming  one  of  the  constants,  as  a  coefficient,  or  an  exponent, 
to  change.  Or  a  secoml  equation  may  be  derived  for  this  part 
of  the  curve  and  one  part  of  the  curve  represented  by  one,  the 
other  by  another  equation.  The  two  equations  may  then  over- 
lap, and  at  some  point  the  curve  represented  equally  well  by 
either  equation,  or  the  ranges  of  application  of  the  two  equa- 
tions may  be  separated  by  a  transition  range,  in  which  neither 
applies  exactly. 

If  neither  the  exponential  functions  nor  the  parabolic  and 
hyperbolic    curves    satisfactorily    represent    the    obser^^ations, 

X 

further  trials  may  be  made  by  calculating  and  tabulating  — 
and  -,  and  plotting  curves  between  x,  y,  -,  -.    Also  expressions 

X  y  X 

as  x^-\-hy^,  and  {x—ay  +  h{y—cy,  etc.,  may  be  studied. 

Theoretical  reasoning  based  on  the  nature  of  the  phenomenon 
represented  by  the  numerical  data  frequently  gives  an  indi- 
cation of  the  form  of  the  equation,  which  is  to  be  expected, 
and  inversely,  after  a  mathematical  equation  has  been  derived 
a  trial  may  be  made  to  relate  the  equation  to  known  laws  and 
thereby  reduce  it  to  a  rational  equation. 

In  general,  the  resolution  of  empirical  data  into  a  mathe- 
matical expression  largely  depends  on  trial,  directed  by  judg- 
ment based  on  the  shape  of  the  curve  and  on  a  knowledge  of 
the  curve  shapes  of  various  functions,  and  only  general  rules 
can  thus  be  given. 

A  number  of  examples  may  illustrate  the  general  methods  of 
reduction  of  empirical  data  into  mathematical  functions. 

153.  Example  1.  In  a  118- volt  tungsten  filament  incan- 
descent lamp,  corresponding  values  of  the  terminal  voltage  e 
and  the  current  i  are  observed,  that  is,  the  so-called  "  volt- 
ampere  characteristic"  is  taken,  and  therefrom  an  equation  for 
the  volt-ampere  characteristic  is  to  be  found. 

The  corresponding  values  of  e  and  i  are  tabulated  in  the 
first  two  columns   of   Table  III  and  plotted  as  curve  I  in 


236 


ENGINEERING  MATHEMATICS. 


Fig.  79.     In  the  third  and  fourth  column  of  Table  III  are 
given  log  e  and  log  i.     In  Fig.  79  then  are  plotted  log  e,  i,  as 
curve  II;  e,  log  i,  as  curve  III;  log  e,  log  i,  a3  curve  IV. 
As  seen  from  Fig.  79,  curve  IV  is  a  straight  line,  that  is 

0.2      0.4      0.6      0.8      1.0      1.2      1.4      1.6      1.8      2.0      2.2      iA=log  e 

15o    2^0=  e 


Fig.   79.     Investigation  of  Volt-ampere  Characteristic   of  Tungsten  Lamp 

Filament. 

log  i  =  A-\-n  log  e;    or    t  =  oe'*, 
which  is  a  parabolic  curve. 

The  constants  a  and  n  may  now  be  calculated  from 
the  numerical  data  of  Table  III  by  the  method  of  leasi 
squares,  as  discussed  in  Chapter  IV,  paragraph  120.  While 
this  method  gives  the  most  accurate  results,  it  is  so  laborious 
as  to  be  seldom  used  in  engineering;  generally,  values  of  the 
constants  a  and  n,  sufficiently  accurate  for  most  practical 


EMPIRICAL  CURVES. 


237 


purposes,  are  derived  by  the  so-called  "  2A  method,"  which, 
with  proper  tabular  arrangement  of  the  numerical  values,  gives 
high  accuracy  with  a  minimum  of  work. 

Table  III. 

VOLT-AMPERE  CHARACTERISTIC  OF  118-VOLT  TUNGSTEN  LAMP. 


e 

i 

log  e 

log  i 

8-211 +0-6  log  f- 

J 

2 

00245 

0-301 

8-392 

8-389 

-0-003 

4 

0  037 

0   602 

8-568 

g.572 

-0004 

8 

0  0568 

0-903 

8-754 

8-753 

+  0-001 

16 

00855 

1-204 

8-932 

8-933 

-0-001 

25 

0-1125 

1-398 

9-051 

9  050 

+  0-001 

32 

0-1295 

1-505 

9-112 

9-114 

-0-002 

50 
64 

0-1715 
0-200 

1.699 
1.806 

9.234 
9.301 

9-230 
9-295 

+  0-004 
+  0-006 

100 

0-2605 

2-000 

9-416 

9-411 

+  0-005 

125 

0-2965 

2097 

9-472 

9-469 

+  0-003 

150 

0-3295 

2176 

9-518 

9-518 

0 

180 

03635 

2-255 

9-561 

9-564 

-0-003 

200 

0-3865 

2-301 

9-587 

5-592 

-0-005 

218 

0407 

2- 338 

9-610 

9-614 

-0-004 

i"7=    7-612 

2-043 

avg.  ±0-003 

I' 

r=  14-973 

6-465 

4-  7  per  cent 

»=     7-361 

4.422 

4-422 

n=    = 

7-361 

0.6007~0 

6 

^■14=  22.585 

8  505 

0-6X22-585      = 

=      13   551 

551  =  4.954 

i  =  8   505-13 

4-954-^14  = 

8-211 

logi=8-211+0  6l 

og  e     and     i  = 

0-016256°' 

The  fourteen  sets  of  observations  are  divided  into  two 
groups  of  seven  each,  and  the  sums  of  log  e  and  log  i  formed. 
They  arc  indicated  as  27  in  Table  III. 

Then  subtracting  the  two  groups  117  from  each  other, 
eliminates  A,  and  dividing  the  two  differences  A,  gives  the 
exponent,  n  =  0.6011;  this  is  so  near  to  0.6  that  it  is  reasonable 
to  assume  that  7i=0.6,  and  this  value  then  is  used. 

Now  the  sum  of  all  the  values  of  log  e  is  formed,  given  as 
S14  in  Table  III,  and  multiplied  with  n  =0.6,  and  the  product 


238  ENGINEERING  MATHEMATICS 

subtracted  from  the  sum  of  all  the  log  i.     The  difference  i 
then  equals  14/1,  and,  divided  by  14,  gives 

A  =  loga  =  8.211; 

hence,  a  =-0.01025,  and  the  volt-ampere  characteristic  of  this 
tungsten  lamp  thus  follows  the  equation, 

log  1  =  8.211 +0.0  log  e; 
i  =  0.01025eO-«. 

From  c  and  i  can  be  derived  the  i)ower  input  p=--ei,  and  the 

e 
resistance  r  =  — : 
I 

p  =  0.01G25ei-6; 

0-4 

e 
r  = 


0.01625' 

and,  eliminating  e  from  these  two  expressions,  gives 

p  =  0.01G25V  =  11.35r4x  10-10, 

that  is,  the  power  input  varies  with  the  fourth  power  of  tlie 
resistance. 

Assuming  the  resistance  r  as  proportional  to  the  absolute 
temperature  T,  and  considering  that  the  power  input  into  the 
lamp  is  radiated  from  it,  that  is,  is  the  power  of  radiation  P,., 
the  equation  between  p  and  r  also  is  the  equation  between  P^ 
and  T,  thus, 

that  is,  the  radiation  is  proportional  to  the  fourth  power  of  the 
absolute  temperature.  This  is  the  law  of  black  body  radiation, 
and  above  equation  of  the  volt-ampere  characteristic  of  the 
tungsten  lamp  thus  appears  as  a  conclusion  from  the  radiation 
law,  that  is,  as  a  rational  equation. 

154.  Example  2.  In  a  magnetite  arc,  at  constant  arc  length, 
the  voltage  consumed  by  the  arc,  e,  is  observed  for  different 
values  of  current  i.  To  find  the  equation  of  the  volt-ampere 
characteristic  of  the  magnetite  arc  : 


EMPIRICAL  CURVES. 


239 


Table  IV. 

VOLT-AMPERE  CHARACTERISTIC  OF  MAGNETITE  ARC. 


t 

e 

log  i 

log  e 

(e-40) 

log  (e-40) 

(e-30) 

log  (e-30) 

ec 

J 

05 

180 

9-699 

2-204 

120 

2. 079 

130 

2. 114 

158 

-2 

1 

120 

0  000 

2-079 

80 

1-903 

90 

1954 

120.4 

+  04 

2 

94 

0301 

1-973 

54 

1732 

64 

1806 

94 

0 

4 

75 

0-602 

1-875 

35 

1544 

45 

1653 

75.  2 

+  02 

8 

62 

0903 

1-792 

22 

1342 

32 

1   505 

62 

0 

12 

56 

1-079 

1-748 

16 

1.204 

26 

1415 

56. 2 

+  02 

-3  = 

=0  000- ■ 

fi    R7J. 

-''3  = 

=2.584- 

4-573 

J  = 

=  2.584.. 

-1-30 

n= 

2-584 

1 

-  =  -0.5034 

^-05 

- -1-301 

i'e=2 

•58". •    - 

10. 447 
-1.292 

2-584X-0. 

5 

A  = 

11-739 

11. 

739-!- 6  = 

1-956  = 

log( 

«-30)  =  1.956-C 
e-30  =9041-" 

)  ■  5  log  i 
5     and     p= 

., ,     90.4 
30 -t-     ._ 

The  first  four  columns  of  Table  IV  give  i,  e,  log  i,  log  e. 
Fig.  80  gives  the  curves:  i,  e,  as  I;  i,  log  e,  as  II;  log  2,  e,  as 
III;  log  i,  log  e,  as  IV. 

Neither  of  these  curves  is  a  straight  line.  Curve  IV  is 
relatively  the  straightest,  especially  for  high  values  of  e.  This 
points  toward  the  existence  of  a  constant  term.  The  existence 
of  a  constant  term  in  the  arc  voltage,  the  so-called  "  counter 
e.m.f.  of  the  arc  "  is  physically  probable.  In  Table  IV  thus 
are  given  the  values  (e— 40)  and  log  (e  — 40),  and  plotted  as 
curve  V.  This  shows  the  opi)osite  curvature  of  IV.  Thus  the 
constant  term  is  less  than  40.  Estimating  by  interjoolation,  and 
calculating  in  Table  IV  (e— 30)  and  log  (e— 30),  the  latter, 
plotted  against  log  i  gives  the  straight  line  VI.  The  curve  law 
thus  is 

log  (e  -  30)  =  il  +  w  log  i. 


240 


ENGINEERING  MATHEMATICS. 


Proceeding  in  Table  IV  in  the  same  manner  with  log  i 
and  log  (e  — 30)  as  was  done  in  Table  III  with  log  e  and  log  i, 
gives 

n=— 0.5;     A  =  loga  =  1.956;     and     a  =  90.4; 


Fig.  80.     Investigation  of  \'olt-ampere  Characteristic  of  Magnetite  Arc. 

hence 

log  (e- 30)  =  1.956-0.5  log  i; 

e-30  =  90.4i-o-5; 


EMPIRICAL  CURVES. 


241 


which  is  the  equation  of  the  magnetite  arc  volt-ampere  charac- 
teristic. 

155.  Example  3.  The  change  of  current  resulting  from  a 
change  of  the  conditions  of  an  electric  circuit  containing  resist- 
ance, inductance,  and  capacity  is  recorded  by  oscillograph  and 
gives  the  curve  reproduced  as  I  in  Fig.  81.     From  this  curve 


\ 

log 
0.-5- 

\ 

,_^ 

L\ 

A^ 

-^ 

"~" 

^ 

N 

i 

A 

\ 

\ 

s 

/^ 

1 

\ 

\ 

N 

k 

\ 

\ 

\ 

\ 

\ 

\ 

\ 

k 

] 

T 

\ 

s 

\ 
II 

\, 

J 

Ve- 

\, 

\ 

\ 

s 

\, 

\ 

K 

IIl\ 

\ 

\ 

V 

\ 

\ 

\ 

\, 

\ 

\ 

I\ 

<.■ 

\ 

^ 

\ 

^^ 

0 

4 

0 

8 

1 

2 

t 

1 

6 

2 

0 

2 

4 

2 

8 

Fig.  81.     Investigation  of  Curve  of  Current  Change  in  Electric  Circuit. 

are  taken  the  numerical  values  tabulated  iis  t  and  %  in  the  first 
two  columns  of  Table  \.  In  the  third  and  fourth  columns  are 
given  log  t  and  log  i,  and  curves  then  plotted  in  the  usual 
manner.  Of  these  curves  only  the  one  between  t  and  log  i 
is  shown,  as  II  in  Fig.  81,  since  it  gives  a  straight  line  for  the 
higher  values  of  /.     For  the  liigher  values  of  t,  therefore, 

logi  =  ^4  — nt;    or,     i  =  a£~"'; 
that  is,  it  is  an  exjjonential  function. 


242 


ENGINEER  ING   MA  THEM  A  TICS. 


Table  V. 
TRANSIENT  CURRENT  CHARACTERISTICS. 


t 

I 

logt 

log  t 

u 

i' 

t          log  i' 

M 

ic 

J 

0 

2.10 

— 

0.322' 

4.94 

2.84 

0 

0.461 

2.85 

2.09 

-0.01 

0.1 

0.2 

2.48 
2.66 

9.000 
9.301 

0.394 
0.425 

4.44 
3.98 

1.96 
1.32 

0.1 

0.292 

1.94 

1.32 

2.50 
2.66 

+  0.02 
0 

0.2 

0.121 

0.4 
0.8 
1-2 

2.58 
2.00 
1.36 

9. 602 
9.903 
0.079 

0.412 
0.301 

3.21 
2.09 
1.36 

0.63 
0.09 
0 

0.4 

9.799 

0.61 
0.13 
0.03 

2.60 
1.96 
1.33 

-1-0.02 
-0.04 
-0.03 

0-8 

8.954 

0-134 

1.6 

0.90 

0.204 

9.954 

0.89 

-0.01 

— 

— 

0.01 

0.88 

-0.02 

2.0 

0.58 
0.34 

0.301 
0.398 

9.763 
9.531 

0.58 
0.34 

0 
0 

0.58 
0.34 

0 
0 

2.5 

3.0 

0.20 

0.477 

9.301 

0.20 

0 

— 

— 

0.20 

0 

^3=       4.8 

9.851 

I2 

-0.1    0.753 

^2=      5.5 

9.851     _ 
„      -9.950 
3 

9.832 

J 

-0.6     9.920 

=  0.5-0.833 

5.5 

—  =2.75 
2 

9.832     _ 
2      -'■'"' 

log 

£X0.5  =  0.217 

^=1.15 
log  eX1.15  = 

-0.534 
0.499 

n 

-0.833 

3.84 

0.217 

ni=  — 

0.534 

=  -1.07 

0.499 

J 

,   =   0.7     0.673 

^■5=      10.3 

8.683 

712  log 

£X0.7  = 

=  -1.167 

1 .840 

10.3  Xm  log 

J  = 

-4.78 

4 
57 

1.840-5-4  =  0.460  =  ^2— log  as 
o?=2.85 

3.4( 

3. 467-5  =  0. 693  =  .4,  =  logm 

log  t2  =  0  .460  —  3  .84<  log  e 

a,  =  < 

L.94 

12=2  .85  =  - 3S4< 

log  tl  = 

3.693-1.07<log  e 

ii  =  4.94£~i''"' 

tc  =  4.94t- 

-  i.07r_ 

2.85.-'-'*' 

To  calculate  the  constants  a  and  n,  the  range  of  values  is 
used,  in  which  the  curve  II  is  straight;  that  is,  from  t  =  1.2 
to  t  =  3.  As  these  are  five  observations,  they  are  grouped  in  two 
pairs,  the  first  3,  and  the  last  2,  and  then  for  t  and  log  i,  one- 
third  of  the  sum  of  the  first  3,  and  one-half  of  the  sum  of  the 
last  2  are  taken.     Subtracting,  this  gives, 

J^  =  1.15;    i  log  1= -0.534. 

Since,  however,  the  equation,  i  =  ae~''\  when  logarithmated, 
gives 

log  {  =  log  a—  ni  log  e, 
thus  i  log  i=—n  log  £  J  t. 


EMPIRICAL  CURVES.  243 

it  is  necessary  to  multiply  At  by  log  £  =  0.4343  before  dividing  it 
into  log  i  to  derive  the  value  of  n.    This  gives  n  =  1.07. 

Taking  now  the  sum  of  all  the  five  values  of  t,  multiplying  it 
by  log  £,  and  subtracting  this  from  the  sum  of  all  the  five  valuer 
of  log  i,  gives  5.4  =  3.467;   hence 

^  =  log  a  =  0.693, 
a  =  4.94, 

and  log  fi  =0.693  -1.07/  log  e; 

The  current  ii  is  calculated  and  given  in  the  fifth  column 
of  Table  V,  and  the  difference  i'  =  J  =  ix—i  in  the  sixth 
column.  As  seen,  from  t  =  \.2  upward,  i\  agrees  with  the 
observations.  Below  ^  =  1.2,  however,  a  difference  i'  remains, 
and  becomes  considerable  for  low  values  of  t.  This  difference 
apparently  is  due  to  a  second  term,  which  vanishes  for  higher 
values  of  t.  Thus,  the  same  method  is  now  applied  to  the 
term  i';  column  8  gives  log  i',  and  in  curve  III  of  Fig.  81  is 
plotted  log  i'  against  t.  This  curve  is  seen  to  be  a  straight 
line,  that  is,  i'  is  an  exponential  function  of  t. 

Resolving  i'  in  the  same  manner,  by  using  the  first  four 
points  of  the  curve,  from  ^  =  0  to  ^  =  0.4,  gives 

log  i2  =  0.460 -3.84nog  e; 
i2  =  2.85£-3-84«. 

and,  therefore, 

t  =  ti_{2  =  4.94£-io7'-2.85£-3-8« 

is  the  equation  representing  the  current  change. 

The  numerical  values  are  calculated  from  this  equation 
and  given  under  %  in  Table  V,  the  amount  of  their  difference 
from  the  observed  values  are  given  in  the  last  column  of  this 
table. 

A  still  greater  approximation  may  be  secured  by  adding 
the  calculated  values  of  22  to  the  observed  values  of  i  in  the 
last  five  observations,  and  from  the  result  derive  a  second 
approximation  of  i\,  and  by  means  of  this  a  second  approxi- 
mation of  l2. 


244 


ENGINEERING  MATHEMATICS. 


156.  As  further  example  may  be  considered  the  resolution 
of  the  core  loss  curve  of  an  electric  motor,  which  had  been 
expressed  irrationally  by  a  potential  series  in  paragraph  144 
and  Table  I. 

Table  VI. 

CORE  LOSS   CURVE. 


e 
Volts. 

Pi  kw. 

log  c 

loK  Pi 

16  log  e 

.4=logPi 
-1-6  log  e 

Pc 

J 

40 

0-63 

1602 

9-799 

2-563 

7-236 

0-70 

-0-07 

60 

1 

36 

1-778 

0-134 

2 

845 

7 

289  1 

293  !     avg. 

1 

34 

+  0 

02 

80 

2 

18 

1.903 

0.338 

3 

045 

7 

2 

12 

+  0 

06 

100 

3 

00 

2.000 

0.477 

3 

200 

7 

277  I  7-282 

3 

03 

-0 

03 

120 

3 

93 

2.079 

0.594 

3 

326 

7 

268  J 

4 

05 

-0 

12 

140 

6 

22 

2.146 

0.794 

3 

434 

7 

360 

5 

20 

+  1 

02 

160 

8 

59 

2.  204 

0-934 

3 

526 

7 

408 

6 

43 

+  2 

16 

y. 

=  5-283 

0-271 

logP 

i  =  7.282  +  1.6loge 

^3-3 

=  1-761 

0-090 

P 

t=l-914e'',  in  watts 

Vr 

=  4-079 

1-071 

i-2-2 

=  2- 0395 

0-535 

'    A 

=  02785 

0-445 

r 

_0-445 
02785 

=  l-598~ 

1-6 

The  first  two  columns  of  Table  VI  give  the  observed  values 
of  the  voltage  e  and  the  core  loss  Pi  in  kilowatts.  The  next 
two  columns  give  log  e  and  log  P,-.  Plotting  the  curves  shows 
that  log  e,  log  Pi  is  approximately  a  straight  line,  as  seen  in 
Fig.  82,  with  the  exception  of  the  two  highest  points  of  the 
curve. 

Excluding  therefore  the  last  two  points,  the  first  five  obser- 
vations give  a  parabolic  curve. 

The  exponent  of  this  curve  is  found  by  Table  VI  as 
71=1.598;  that  is,  with  sufficient  approximation,  as  n  =  l.(). 

To  see  how  far  the  observations  agree  with  the  curve,  as 
given  by  the  equation, 


in  the  fifth  column  1.6  log  e  is  recorded,  and  in  the  sixth  column, 
A  =  log  a  =  log  Pi— 1.6  log  c.  As  seen,  the  first  and  the  last 
two  values  of  A  differ  from  the  rest.     The  first  value  corre- 


EMPIRICAL  CURVES. 


245 


spends  to  such  a  low  value  of  Pi  as  to  lower  the  accuracy  of 
the  observation.  Averaging  then  the  four  middle  values, 
gives  A  =7.282;  hence, 

IogPi  =  7.282  +  l.Gloge, 
Pi  =  1.914ei-6'  in  watts. 


1.6 

1 

7 

1.8 

1.9 

2.0 

2.1 

2.2 

loq 

Pi 

lo 

ge 

1 

{ 

P. 

/ 

kw. 

/ 

(S 

A 

/ 

y 

y 

,<?\ 

X 

r 

?0  ^ 

r 

\4\ 

y 

< 

( 

/ 

,y 

/ 

/ 

AU 

/ 

/ 

/ 

-5;0 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

) 

<! 

r 

( 

y' 

y 

y 

J 

< 

J 

0 

i 

a 

6 

) 

8 

) 

e=Vc 
i5o 

)ltS 

li 

0 

1 

10 

1( 

Q 

Fig.  82.     Investigation  of  Cuvres. 


This  equation  is  calculated,  as  Pc,  and  plotted  in  Fig.  82. 
The  observed  values  of  Pi  are  marked  by  circles.  As  seen, 
the  agreement  is  satisfactory,  with  the  exception  of  the  two 
highest  values,  at  which  apparently  an  additional  loss  appears, 
which  does  not  exist  at  lower  voltages.  This  loss  probably  is 
due  to  eddy  currents  caused  by  the  increasing  magnetic  stray 
field  resulting  from  magnetic  saturation. 


246 


ENGINEERING  MATHEMATICS. 


157.  As  a  further  exami)le  may  be  considered  the  resolution 
of  the  magnetic  characteristic,  plotted  as  curve  I  in  Fig.  83, 
and  given  in  the  fii'st  two  columns  of  Table  VII  as  H  and  B. 


Table  VII. 
MAGNETIC  CHARACTERISTIC. 


H 

B 

kilolines 

log  H 

log  B 

B 

H 

H 
B 

Be 

J 

2 

30 

0301 

0477 

1-5 

0-667 

6-4 

+  3-4 

4 

8 

4 

0 

602 

0 

924 

2 

1 

0476 

9 

7 

+  1-3 

6 
8 

11 
13 

2 
0 

0 
0 

778 
903 

049 

114 

1 
1 

867 
625 

0  536 
0  614 

11 
13 

6 
0 

+  0-4 
0 

10 

14 

0 

000 

146 

1 

40 

0715 

13 

9 

-0-1 

15 

15 

4 

176 

188 

1 

033 

0  974 

15 

45 

+  0-05 

20 

16 

3 

301 

212 

0 

815 

1-227 

16 

3 

0 

30 

17 

2 

477 

236 

0 

573 

1-74 

17 

3 

+  01 

40 

17 

8 

602 

250 

0 

445 

2-25 

17 

8 

0 

60 

18 

5 

778 

267 

0 

308 

3-25 

18 

4 

-01 

80 

18 

8 

903 

274 

0 

235 

4-25 

18   8 

0 

-£■4  =  53 

3-530 

^"4  =  210 

11-49 

i  =  157 

7  96 

=  00507  =& 

157 

7-96 

i-8  =  283 

263X0  0507  = 

15020 
=13  334 

=    1-686 

1.686-8  = 

=   0211  = 

a 

H 

).211+0  0507^"     and     B=  — 
0  2 

H 

11  +  0  0507 

ll' 

Plotting  H,  B,  log  H,  log  B  against  each  other  leads  to  no 
results,  neither  does  the  introduction  of  a  constant  term  do 
this.     Thus  in  the  fifth  and  sixth  columns  of  Table  VII  are 

B  77 

calculated  -rz  and  -"-,  and  are  plotted  against  H  and  against  B. 
H  ti 

Of  these  four  curves,  only  the  curve  of  -g  against  H  is  shown 

in  Fig.  S3,  as  II.     This  curve  is  a  straight  line  with  the  exception 
of  the  lowest  values;  that  is, 

H 


B 


=  a-\-hH. 


EMPIRICAL  CURVES. 


247 


Excluding  the  three  lowest  values  of  the  observations,  as 
not  lying  on  the  straight  line,  from  the  remaining  eight  values, 
as  calculated  in  Table  VII,  the  following  relation  is  derived. 


=  0.211 +0.0507  fi, 


r 

/■ 

/■ 

/ 

/ll 

/ 

/ 

B 

/ 

^^ 

y 

I 

/" 

^' 

/ 

/ 

c 

/ 

/ 

/ 

<u 

c 

/ 

r 

/ 

r 

/ 

'' 

r 

1 

] 

2 

0 

i 

0 

4 

0    H    . 

[) 

6 

!) 

7 

) 

80 

Fig.  83.     Investigation  of  Magnetization  Curve. 

and  herefrom, 

H 


B 


0.211 +0.0507  H 


is  the  equation  of  the  magnetic  characteristic  for  values  of  H 
from  eight  upward. 

The  values  calculated  from  this  equation  are  given  as  B^ 
in  Table  VII 


248  ENGINEERING  MATHEMATICS. 

H 

The  difference  between  the  observed  values  of  ~,  and  the 

B 

value  given  by  above  equation,  which  is  appreciable  up  to 

H=Q,  could  now  be  further  investigated,  and  would  be  found 

to  approximately  follow  an  exponential  law. 

As  a  final  example  may  be  considered  the  investigation  of 
a  hysteresis  curve  of  silicon  steel,  of  which  the  numerical  values 
are  given  in  columns  1  and  2  of  Table  VIII. 

The  first  column  gives  the  magnetic  density  B,  in  Hues  of 
magnetic  force  per  cm.^;  the  second  column  the  hysteresis  loss 
10,  in  ergs  per  cycle  per  kg.  (specific  density  7.5).  The  third 
column  gives  log  B,  and  the  fourth  column  log  iv. 

Of  the  four  curves  between  B,  w,  log  B,  log  iv,  only  the 
curve  relating  log  iv  to  log  B  approximates  a  straight  line,  and 
is  given  in  the  upper  part  of  Fig.  84.  This  curve  is  not  a 
straight  line  throughout  its  entire  length,  but  only  two  sections 
of  it  are  straight,  from  B  =  50  to  5  =  400,  and  from  5  =  1600  to 
5=8000,  but  the  curve  bends  between  500  and  1200,  and  above 
8000. 

Thus  two  empirical  formulas,  of  the  form:  iv  =  aB",  are 
calculated,  in  the  usual  mamier,  in  Table  VIII.  The  one 
applies  for  lower  densities,  the  other  for  medium  densities: 

Low  density:  5^400:  iv  =  0.00S4lB^'^^ 

Medium  density :  1600  ^  5 ^  8000 :      iv  =  0.1096B^-^ 

In  Table  VIII  the  values  for  the  lower  range  are  denoted 
by  the  index  1,  for  the  higher  range  by  the  index  2. 

Neither  of  these  empirical  formulas  applies  strictly  to  the 
range:  400 <B<  1600,  and  to  the  range  5 > 8000.  They  may 
be  applied  within  these  ranges,  b}^  assuming  either  the  coefficient 
a  as  varying,  or  the  exponent  n  as  varying,  that  is,  applying  a 
correction  factor  to  a,  or  to  n. 

Thus,  in  the  range:  400 <5<  1600,  the  loss  may  be  repre- 
sented by: 

(1)  An  extension  of  the  low  density  formula: 

wj  =  ai52-ii      or      u'  =  0.003415".. 

(2)  An  extension  of  the  medium  density  formula 

w  =  a<>B^-^       or      w  =  0.109GB"i, 


EMPIRICAL  CURVES. 


249 


by  giving  tables  or  curves  of  a  respectively  n.     Such  tables  are 
most  conveniently  given  as  a  percentage  correction. 


] 

p) 

V 

Log 

w 

,^ 

^ 

^ 

r^ 

y 

-6 

,/« 

y 

> 

i^ 

f^ 

^ 

_,A 

Y 

-4 

^ 

f 

y 

^ 

*/ 

^/ 

> 

^. 

id 

<> 

,;a 

/^ 

/ 

^ 

' 

-2 

y^ 

! 

0- 

^ 

i 

/ 

0- 

/ 

— 1 

J 

V 

B- 

y 

/ 

/ 

0- 

/ 

y 

- 

^ 

/ 

^ 

U 

^ 

^ 

<" 

/ 

h 

'  1 

V, 

s. 

/ 

/ 

)— 

a, 

^•7 

y 

/ 

y 

/^' 

\ 

' 

)- 

/ 

/ 

\, 

-2 

4 

/ 

> 

\ 

j 

-2 

A- 

l—ri 

\ — 1 

1—k 

^-, 

I 

1  / 

-2 

n 

^ 

-N 

^-•1 

-n 

V 

-1 

X 

N 

s 

V 

-1 

S 

Sj 

-< 

/ 

a>- 

< 

^ 

— 

*-^ 

^ 

' 

'     ' 

'» 

-I 

LogB  = 

2X, 

3X) 

1 

4X, 

Fig.  84. 


The  percentage  correction,  which  is  to  be  applied  to  ai  and 
02  respectively,  to  rii  and  n2,  to  make  the  formulas  applicable 


250 


ENGINEERING   MA  THEM  A  TICS. 


Table  VIII. 

HYSTERESIS  OF  SILICON  STEEL. 


Jai 

Jai 

Jni 

inj 

B 

w 

log  5 

logu) 

ai 

at 

m 

m 

m 

m 

m 

7o 

% 

% 

% 

35 
50 

6.4 
13 

1.544 

0.806 

+   3.0 

— 

+  0.46 

— 

2.120 

— 

2.03 

1.699 

1.117 

-0.2 

-0.03 

2.109 

60 

19 

1.778 

1.279 

-    1.1 

— 

-0.13 

— 

2.107 

— 

2.14 

80 

36 

1  .903 

1  .556 

+    1.9 

— 

+  0.20 

— 

2.114 

— ■ 

2.12 

100 

57 

2.000 

1  .752 

-0.2 

— 

-0.02 

— 

2.110 

— 

2.09 

120 
160 

83 

156 

2.079 

1  .922 

+   0.7 
+   2.3 

+  0.07 
+  0.21 

2.111 
2.114 

2.16 
2.10 

2.204 

2.193 

200 

245 

2.301 

2.389 

+    0.2 

— 

+  0.02 

— 

2.110 

— 

2.07 

250 

394 

2.398 

2.595 

+    0.7 

— 

+  0.06 

— 

2.111 

— 

2.03 

300 

571 

2.477 

2.757 

-0.5 

— 

-0.04 

— 

2.109 

— 

1.98 

400 
500 

1025 
1610 

2.602 

3.011 

-2.7 

-29.5 

-0.22 
-0.37 

-3.52 

2.105 

1.544 

2.03 
2.02 

2.699 

3.207 

-    4.7 

2.102 

600 

2320 

2.778 

3  .366 

-6.2 

-24.0 

-0.87 

-2.68 

2.092 

1.557 

1.96 

800 

4030 

2.903 

3  .605 

-16.5 

-15.8 

-1.17 

-1.72 

2.085 

1.573 

1.91 

1000 

6150 

3.000 

3.789 

-15.7 

-11.1 

-1.14'-1.06 

2.086 

1.583 

1.89 

1200 
1600 

8680 
14370 

3.079 

3.938 

-18.7 
-26.9 

-6.0 

-2.02 

-0.55 

2.067 

1  .5912 

1.82 
1.73 

3  .204 

4.157 

-2.0 

-0.18 

1.5971 

2000 

21000 

3.301 

4.322 

— 

0.0 

— 

0.00 

— 

1  .6000 

1.67 

2500 

30300 

3.398 

4.481 

— ■ 

+    0.9 

— 

+  0.07 

— 

1.6011 

1.62 

3000 
4000 

40500 
63400 

3  .477 

4.607 

I 

+    1.2 
-0.2 

+  0.09 

1.6014 
1.5997 

1.58 
1.58 

3.602 

4.802 

-0.02 

5000 

90600 

3.699 

4. 957 

— 

-0.2;     — 

-0.02 

— 

1.5997 

1.59 

6000 

120600 

3  .778 

5.082 

— 

-    0.9 

— 

-0.07 

— 

1.5989 

1.61 

8000 
10000 

194100 
282500 

3  .903 

5.288 

+    0.7 

+  0.05 

1  .6008 

1.66 
1.77 

4.000 

5.451 

+    2.6 

+  0.17 

1  .6027 

12000 

397500 

4.079 

5. 599 

— 

+    7.9 

— 

+  0.50 

— 

1  .6080 

2.32 

14000 

609500 

4.146 

5. 785 

—     1+27.7 

— 

+  1.67 

— • 

1.6267 

2.88 

16000 

907500 

4.204 

5   958 

—       +45.9!     — 

-1-2.85 

— 

1  .6456 

— 

^i=    9.459 

7.626 

J  = 

11.982 

12.945 

2.523 

5.319 

_  5jn9 

=    2.11 

"'        2.523 

-^"10  =  21  .441 

20.571 

2.11X21.441  = 

=45.241 

J=- 

-24.670 

]ogu>=7.533 

^-2.lllo 

gB 

-10  = 

-2.467 
=  7.533 

=  logai                      u;  =  0.0034 

iB'" 

a\ 

=  0.0034 

1 

-I'<  =  13.380 

17.567 

V    _ 
J  = 

14.982 

20.129 

=    1.602 

2.562 

2.562 

"2=~~vr;,= 

=    1  .599 

2^1  .60 

1 .602 

'3  =28.362 

37.696 

1.60X28.362 
J  = 

=  45.379 

log«)  =  9.040-t 

-1.60  lo( 

iB 

—  7.683 

-8  = 

-  0.960 
=  9.040 

=  logai                    «'  =  0.1096j 

5'-' 

as 

=  0.109 

i 

EMPIRICAL  CURVES.  251 

to  the  ranges  where  the  logarithmic  curve  is  not  a  straight 
line,  are  given  in  Table  Xlll  as 

Jai      Ja2      ^n\      Jn-z 
di  '      a-z'      ni  '      n-z' 

they  are  calculated  as  follows : 

Assuming  n  as  constant,  =no,  then  a  is  not  constant,  =ao, 
and  the  ratio: 

Ja    a 
a     ao 

is  the  correction  factor,  and  it  is: 


w  =  aB'">, 

hence : 

log  w  =  log  a +  710  log  B 

and 

log  a  =  log  IV  —  no  log  B; 

thus: 

and 


log  —  =  log  a  —log  Go  =  log  IV  —log  ao  —no  log  B, 
ao 


Ja    a 


= l=A^log  ty- log  ao— nologB— 1.       .     (1) 

a      ao 


Assuming  a  as  constant,  =ao,  then  n  is  not  constant,  =no, 

and  the  ratio, 

Jn     n 

—  = 1, 

n      no 

is  the  correction  factor,  and  it  is 

w  =  aQB''; 
hence 

log  w  =  log  ao  +  n  log  5, 


252  ENGINEERING  MATHEMATICS. 

and 


thus 


and 


n  log  5=  log  w  —  log  ao] 

n       n  log  B     log  w  —log  ao 
no     no  log  B         uq  log  B     ' 


dn_n  \ogw-\ogao-no\ogB 

n      iiQ  no  log  5 

by  these  equations  (1)  and  (2)  the  correction  factors  in  columns 
5  to  8  of  Table  VIII  are  calculated,  by  using  for  ao  and  no  the 
values  of  the  lower  range  curve,  in  columns  5  and  7,  and  the 
values  of  the  medium  range  curve,  in  columns  6  and  8. 

Thus,  for  instance,  at  J5  =  1000,  the  loss  can  be  calculated 
by  the  equation, 

w  =  aiB''\ 

by  applying  to  ax  the  correction  factor: 

—15.7  per  cent  at  constant:  ni  =2.11,  that  is, 

ax  =0.00341(1  -0.157)  =0.00287; 

or  by  applying  to  Ux  the  correction  factor: 

—1.14  per  cent  at  constant:  ai  =0.00341,  that  is, 

ni  =2.11(1 -0.0114)  =2.086. 

Or  the  loss  can  be  calculated  by  the  equation, 

w  =  a2B''\ 

by  applying  to  a2  the  correction  factor: 

—  11.1  per  cent  at  constant:  n2  =  1.60,  that  is, 

02  =  0.1096(1 -0.111)  =0.0974; 


EMPIRICAL  CURVES.  253 

or  by  applying  to  112  the  correction  factor: 

—1.06  per  cent  at  constant:  02  =  0.1096,  that  is, 

n2  =  1 .60(1  -0.0106)  =  1 .583 , 

and  the  loss  may  thus  be  given  by  either  of  the  four  ex- 
pressions : 

i/;  =  0.0028752-^^  =  0.00341B2086  =  0.0974B^-^  =  0.10965i-^«^ 

As  seen,  the  variation  of  the  exponent  n,  required  to  extend 
the  use  of  the  parabolic  equation  into  the  range  for  which  it 
does  not  strictly  apply  any  more,  is  much  less  than  the  varia- 
tion of  the  coefficient  a,  and  a  far  greater  accuracy  is  thus 
secured  by  considering  the  exponent  n  as  constant — 1.6  for 
medium  and  high  values  of  B—  and  making  the  correction  in 
coefficient  a,  outside  of  the  range  where  the  1.6th  power  law 
holds  rigidly. 

In  the  last  column  of  Table  \'III  is  recorded  the  ratio  of 

.     .  ^  log  w  ,  ,      ^  X 

variation,  m  =  — ^^^ — ,  as  the  averages  each  of  two  successive 
J  log  B 

values.  As  seen,  m  agrees  with  the  exponent  n  within  the 
two  ranges,  where  it  is  constant,  but  differs  from  it  outside 
of  these  ranges.  For  instance,  if  B  changes  from  1600  down- 
ward, the  ratio  of  variation  m  increases,  while  the  exponent 
n  slightly  decreases. 

In  Fig.  84  are  shown  the  percentage  correction  of  the 
coefficients  cti  and  02,  and  also  the  two  exponents  Ui  and  n-z, 
together  with  the  ratio  of  variation  w?. 

The  ratio  of  variation  m  is  very  useful  in  calculating  the 
change  of  loss  resulting  from  a  small  change  of  magnetic  density, 
as  the  percentual  change  of  loss  w  is  m  times  the  percentual 
(small)  change  of  density. 

As  further  example,  the  reader  may  reduce  to  empirical 
equations  the  series  of  observations  given  in  Table  IX.  This 
table  gives: 

A.  The  candle-power  L,  as  function  of  the  power  input  p, 
of  a  40-watt  tungsten  filament  incandescent  lamp. 

B.  The  loss  of  power  by  corona  (tlischarge  into  the  air),  p, 
in  kw.,  in  1.895  km.  of  conductor,  as  function  of  the  voltage 
e  (in  kv.)  between  conductor  and  return  conductor,  for  the 


254 


ENGINEERING   MA  THE  MA  TICS. 


Table  IX. 


A.     Luminosity  characteristic  of  40-watt  tungsten  incandescent  lamp. 

L  =  horizontal  candlepower. 

p  =  watts  input. 

L 

P 

L 

P 

L 

P 

L 

P 

L 

P 

2 

12.25 

20 

31.64 

40 

44.14 

128 

76.77 

382 

135.6 

4 

16.33 

24 

34.55 

44 

45.42 

192 

95.24 

460 

145.2 

8 

21.35 

28 

37.29 

48 

47.05 

256 

109.0 

— 

— 

12 

25.60 

32 

39.26 

64 

54.31 

291 

118.2     ' 

— 

— 

16 

28.91 

36 

41.47 

96 

65.73 

320 

123.1 

~ 

~ 

B.     Corona  loss  of  high-voltage  transmission  line;  at  60  cycles: 

1895  m  .  length  of  conductor. 

3  .10  m  .  distance  between  conductors. 

No.  000  seven-strand  cable,  1-18  cm.  diameter. 

—  13°C.;   76  .2  cm.  barometer;  sun.shine. 

e=kilovolt8  between  conductors,  eflfective. 

p  =  kilowatts  loss. 

e 

P 

e 

P 

e 

P 

e 

P 

e 

P 

79.8 

0.01 

141.5 

0.09 

181.0 

1.02 

221.0 

8.70 

212.0 

6.44 

90.7 

0.01 

147.0 

0.08 

186.2 

1.55 

227.0 

10.66 

219.0 

8.31 

101.5 

0.02 

153.6 

0.12 

192.6 

2.49 

234.0 

13.25 

— 

— 

109.5 

0.03 

159.0 

0.16 

200.6 

3.77 

189.0 

2.10 

— 

— 

120.5 

0.04 

169.8 

0.35 

208.6 

5.34 

195.0 

2.88 

— 

— 

130.0 

0.06 

174.0 

0.53 

216.0 

7.13 

203.8 

4.72 

— 

C.     Volu 

me-pressure  characteristic  of  dry  steam  at  its  boiling-poiat. 
<  =  degrees  C. 
P=pressure,  in  kg.  per  cm.^ 
y=volume,  in  m.^  per  kg. 

t 

P 

V 

t 

P 

V 

t 

P 

V 

59  .8 

0.2 

7.806 

132.8 

3.0 

0.612 

169.5 

8.0 

0.244 

80.9 

0.5 

3.297 

142.8 

4.0 

0.467 

178  .9 

10.0 

0.197 

99.1 

1.0 

1.717 

151.0 

5.0 

0.379 

186.9 

12.0 

0.167 

119.6 

2.0 

0.896 

157.9 

6.0 

0.319 

197.2 

15.0 

0.135 

EMPIRICAL  CURVES.  255 

distance  of  310  cm.  between  the  conductors,  and  the  conductor 
diameter  of  1.18  cm. 

C.  The  relation  between  steam  pressure  P,  in  kg.  per  cm.^, 
and  the  steam  volume  V,  in  m.^,  at  the  boiling-point,  per  kg. 
of  dry  steam. 

D.  Periodic  Curves. 

158.  All  periodic  functions  of  time  or  distance  can  be  ex- 
pressed by  a  trigonometric  series,  or  Fourier  series,  as  has  been 
discussed  in  Chapter  III,  and  the  methods  of  resolution,  and 
the  arrangements  to  carry  out  the  work  rapidly,  have  also 
been  discussed  in  Chapter  III. 

The  resolution  of  a  periodic  function  thus  consists  in  the 
determination  of  the  higher  harmonics,  which  are  superimposed 
on  the  fundamental  wave. 

As  periodic  functions  are  of  the  greatest  importance  in  elec- 
trical engineering,  in  the  theory  of  alternating  current  pheno- 
mena, a  familiarity  with  the  wave  shapes  produced  by  the  dif- 
ferent harmonics  is  desirable.  This  familiarity  should  be 
sufficient  to  enable  one  in  most  cases  to  judge  immediately  from 
the  shape  of  the  wave,  as  given  by  oscillograph,  etc.,  on  the  har- 
monics which  are  present  or  at  least  which  predominate. 

The  effect  of  the  lower  harmonics,  such  as  the  third,  fifth, 
etc.,  (or  the  second,  fourth,  etc.,  where  present),  is  to  change 
the  shape  of  the  wave,  make  it  differ  from  sine  shape,  giving 
such  features  as  flat  top  wave,  peaked  wave,  saw-tooth,  double 
and  triple  peaked,  steep  zero,  fiat  zero,  etc.,  while  the  high 
harmonics  do  not  change  the  shape  of  the  wave  so  much,  as 
superimpose  ripples  on  it. 

Odd  Lower  Harmonics. 

159.  To  elucidate  the  variation  in  shape  of  the  alternating 
waves  caused  by  various  lower  harmonics,  superimposed  upon 
the  fundamental  at  different  relative  positions,  that  is,  different 
phase  angles,  in  Figs.  85  and  86  are  shown  the  effect  of  a  third 
harmonic,  of  10  per  cent  and  30  per  cent  of  the  fundamental, 
respectively.  A  gives  the  fundamental,  and  C  D  E  F  G  the 
waves  resulting  by  the  superposition  of  the  triple  harmonic 
in  phase  with  the  fundamental  (C),  under  45  deg.  lead  (/)),  90 
deg.  lead  or  quadrature  (E),  135  deg.  lead  (F)  and  opposition 


256 


ENGINEERING  MATHEMATICS. 


(G).  (The  phase  differences  here  are  referred  to  the  maximum 
of  the  fundamental:  with  waves  of  different  frequencies,  the 
phase  differences  natui-ally  change  from  point  to  point,  and  in 
speaking  of  phase  difference,  the  reference  point  on  the  wave 


Third  Harmonic 


Fig.  85.     Effect  of  Small  Third  Harmonic. 

must  thus  be  given.     For  instance,  in  C  the  third  harmonic  is 
in  pliase  with  the  fundamental  at  the  maximum  point  of  the 
latter,  but  in  opposition  at  its  zero  point.) 
The  equations  of  these  waves  are: 

A:  i/ =100  cos/? 
C:  y=100cos^  +  10cos3/3 
E:  y  =  100  cos  /?+10  cos  (3^  +  90  deg.) 
G:  2/ =  100  cos  /3  +  10  cos  (3^+180  deg.) 
=  100cos/?-10cos3^ 


EMPIRICAL  CURVES. 


257 


C:  2/=100cos/3  +  30cos3;5 
D:  y=100  cos  /3  +  30  cos  (3/3  +  45  deg.) 
E:  7/ =100  cos  /?  +  30  cos  (3^3  +  90  deg.) 
F:  2/ =  100  cos  /?  +  30  cos  (3/3  +  135  deg.) 
G:  i/  =  100  cos  /3  +  30  cos  (3/3  +  180  deg.) 
=  100  cos  /3-30  cos  3/3 


Fig.  86.     Effect  of  Large  Third  Harmonic. 


In  all  these  waves,  one  cycle  of  the  triple  harmonic  is  given  in 
dotted  lines,  to  indicate  its  relative  position  and  intensity,  and 
the  maxima  of  the  harmonics  are  indicated  by  the  arrows. 


258 


ENGINEERING  MATHEMATICS. 


As  seen,  with  the  harmonic  in  phase  or  in  opposition  (C  and 
G),  the  waves  are  symmetrical;  with  the  harmonic  out  of  phase, 
the  waves  are  unsymmetrical,  of  the  so-called  "saw  tooth" 
type,  and  the  saw  tooth  is  on  the  rising  side  of  the  wave  with  a 
lagging,  on  the  decreasing  side  with  a  leading  triple  harmonic. 


Third  Harmonic  Flat  Zero  &  Reversal 


Fig.  87.     Flat  Zero  and  Reversal  by  Third  Harmonic. 

The  latter  are  shown  in  D,  E,  F;  the  former  have  the  same  shape 
but  reversed,  that  is,  rising  and  decreasing  side  of  the  wave 
interchanged,  and  therefore  are  not  shown. 

The  triple  harmonic  in  phase  with  the  fundamental,  C,  gives 
a  peaked  wave  with  fiat  zero,  and  the  peak  and  the  flat    zero 


EMPIRICAL  CURVES. 


259 


become  the  more  pronounced,  the  higher  the  third  harmonic, 
until  finally  the  flat  zero  becomes  a  double  reversal  of  volt- 
age, as  shown  in  Fig.  87d. 

Fig.  87  shows  the  effect  of  a  gradual  increase  of  an  in-phase 
triple  harmonic:  a  is  the  fundamental,  b  contains  a  10  per 


5^  Fifth  Harmonic 


Fig.  88.    Effect  of  Small  Fifth  Harmonic. 

cent,  c  a  38.5  per  cent  and  d  a  50  per  cent  triple  harmonic,  as 
given  by  the  equations: 


Qi  y=lQO  cos  /? 

h:  y=100  cos /?+ 10  cos  3^ 

c:  ij  =100  cos  ;9  +  38.5  cos  3/3 

d:  T/  =  100cos/3  +  50cos3^ 


260 


ENGINEERING  MATHEMATICS. 


At  r.  the  wave  is  entirely  horizontal  at  the  zero,  that  is,  remains 
zero  for  an  appreciable  time  at  the  reversal.  In  this  figure,  the 
three  harmonics  are  showTi  separately  in  dotted  lines,  in  their 
relative  intensities. 

A  triple  harmonic  in  opposition  to  the  fundamental  (Figs. 
85  and  86G)  is  characterized  by  a  flat  top  and  steep  zero,  and 


20  ;i  Pifth  Harmonic 


Fig.  89.    Effect  of  Large  Fifth  Harmonic. 


with  the  increase  of  the  third  harmonic,  the  flat  top  develops 
into  a  double  peak  (Fig.  86G),  while  steepness  at  the  point  of 
reversal  increases. 

The  simple  saw  tooth,  produced  by  a  triple  harmonic  in 
quadrature  with  the  fundamental  is  shown  in  Fig.  85E.  AVith 
increasing  triple  harmonic,  the  hump  of  the  saw  tooth  becomes 


EMPIRICAL  CURVES.  261 

more  pronounced  and  changes  to  a  second  and  lower  peak,  as 
shown  in  Fig.  86.  This  figure  gives  the  variation  of  the  saw- 
tooth shape  from  45  to  45  deg.  phase  difference :  With  the  phase 
of  the  third  harmonic  shifting  from  in-phase  to  45  deg.  lead,  the 
flat  zero,  by  moving  up  on  the  wave,  has  formed  a  hump  or  saw 
tooth  low  down  on  the  decreasing  (and  with  45  deg.  lag  on  the 
increasing)  side  of  the  wave.  At  90  deg.  lead,  the  saw  tooth  has 
moved  up  to  the  middle  of  the  down  branch  of  the  wave,  and 
with  135  deg.  lead,  has  moved  still  further  up,  forming  practi- 
cally a  second,  lower  peak.  With  180  deg.  lead — or  opposition 
of  phase — the  hump  of  the  saw  tooth  has  moved  up  to  the 
top,  and  formed  the  second  peak — or  the  flat  top,  with  a  lower 
third  harmonic,  as  in  Fig.  85G. 

Figs.  88  and  89  give  the  effect  of  the  fifth  harmonic,  super- 
imposed on  the  fundamental,  of  5  per  cent  in  Fig.  88,  and  of  20 
per  cent  in  Fig.  89.  Again  A  gives  the  fundamental  sine  wave, 
C  the  effect  of  the  fifth  harmonic  in  opposition  with  the  funda- 
mental, E  in  quadrature  (lagging)  and  G  in  phase.  One  cycle 
of  the  fifth  harmonic  is  shown  in  dotted  lines,  and  the  maxima 
of  the  harmonics  indicated  by  the  arrows. 

The  equations  of  these  waves  are  given  by: 

A:  ?/ =100  cos  ^9 

C:  y  =100  cos  ^-5  cos  5,3 

E:  y  =  100  cos  ^-5  cos  (5^9  +  90  deg.) 

G:  y=100  cosy5  +  5  cos  5,3 

A:  y=100  cos /? 

C:  y  =  100  cos /9- 20  cos  5^ 

E:  y=100  cos  ^-20  cos  (5^  +  90  deg.) 

G:  y=100  cos  .5  +  20  cos  5.3 

In  the  distortion  caused  by  the  fifth  harmonic  (in  opposi- 
tion to  the  fundamental)  flat  top  (Fig.  88C)  or  double  peak  (at 
higher  values  of  the  harmonic,  Fig.  89C),  is  accompanied  by  flat 
zero  (or,  at  very  high  values  of  the  fifth  harmonic,  double  rever- 
sal at  the  zero,  similar  as  in  Fig.  87rf),  while  in  the  distortion 
by  the  third  harmonic  it  is  accompanied  by  sharp  zero. 

With  the  fifth  harmonic  in  phase  with  the  fundamental,  a 
peaked  wave  results  with  steep  zero.  Fig.  88(7.  and  the  transi- 


262 


ENGINEERING  MATHEMATICS. 


tion  from  the  steep  zero  to  the  peak,  with  larger  values  of  the 
fifth  harmonic,  then  develops  into  two  additional  peaks,  thus 
gi\dng  a  treble  peaked  wave,  Fig.  88(t,  with  steep  zero.  The 
beginning  of  treble  peakedness  is  noticeable  already  in  Fig. 
88(j,  with  only  5  per  cent  of  fifth  harmonic. 


10 ;«  Third  Harmonic  & 
5  Jf   Fifth  Harmonic 


Fig.  90.     Third  and  Fifth  Harmonic. 

With  the  seventh  harmonic,  the  treble-peaked  wave  would 
be  accompanied  by  flat  zero,  and  a  quadruple-peaked  wave 
would  give  steep  zero  (Fig.  95). 

The  fifth  harmonic  out  of  phase  with  the  fundamental  again 
gives  saw-tooth  waves,  Figs.  88  and  S^E,  but  the  saw  tooth 


EMPIRICAL  CURVES. 


263 


produced  by  the  fifth  harmonic  contains  two  humps,  that  is, 
is  double,  with  one  hump  low  down,  and  the  other  high  up  on 
the  curve,  thereby  giving  the  transition  from  the  symmetrical 
double  peak  C  to  the  symmetrical  treble  peak  G. 


10^  Third  Harmonic  & 
5  /^  Fifth  Harmonic 


Fig.  91.     Third  and  Fifth  Harmonic. 


i6o.  Characteristic  of  the  effect  of  the  third  harmonic 
thus  is: 

Coincidence  of  peak  with  flat  zero  or  double  reversal,  of  steep 
zero  with  flat  top  or  double  peak,  and  single  hump  or  saw  tooth, 


26'i 


ENGINEERING  MATHEMATICS. 


While  characteristic  of  the  effect  of  the  fifth  harmonic  is: 
Coincidence  of  peak  with  steep  zero,  or  treble  peak,  of  flat 

top  or  double  peak  with  flat  zero  or  double  reversal,  and  double 

saw-tooth. 


10  J?  Third  Harmonic  & 
5  '^  Fifth  Harmonic 


Fig.  92.     Third  and  Fifth  Harmonic. 

By  thus  combining  third  and  fifth  harmonics  of  proper 
values,  they  can  be  made  to  neutralize  each  other's  effect  in  any 
one  of  their  characteristics,  but  then  accentuate  each  other  in 
the  other  characteristic. 

Thus  peak  and  flat  zero  of  the  triple  harmonic  combined  with 
peak  and  steep  zero  of  the  fifth  harmonic,  gives  a  peaked  wave 
with  normal  sinusoidal  appearance  at  the  zero  value;  combin- 


EMPIRICAL  CURVES.  265 

ing  the  flat  tops  or  double  peaks  of  both  harmonics,  the  flat 
zero  of  the  one  neutrahzes  the  steep  zero  of  the  other,  and  we 
get  a  flat  top  or  double  peak  with  normal  zero.  Or  by  com- 
bining the  peak  of  the  third  harmonic  with  the  flat  top  of  the 
fifth  we  get  a  wave  with  normal  top,  but  steep  zero,  and  we  get  a 
wave  with  normal  top,  but  flat  zero  or  double  reversal,  by  com- 
bining the  triple  harmonic  peak  with  the  fifth  harmonic  flat  top. 

Thus  any  of  the  characteristics  can  be  produced  separately 
by  the  combination  of  the  third  and  fifth  harmonic. 

By  combining  third  and  fifth  harmonics  out  of  phase  with 
fundamental — such  as  give  single  or  double  saw-tooth  shapes,  the 
various  other  saw-tooth  shapes  are  produced,  and  still  further 
saw-tooth  shapes,  by  combining  a  symmetrical  (in  phase  or  in 
opposition)  third  harmonic  with  an  out  of  phase  fifth,  or 
inversely. 

These  shapes  produced  by  the  superposition,  under  different 
phase  angles,  of  fifth  and  third  harmonics  on  the  fundamental, 
and  their  gradual  change  into  each  other  by  the  shifting  in 
phase  of  one  of  the  harmonics,  are  shown  in  Figs.  90,  91  and  92 
for  a  third  harmonic  of  10  per  cent,  and  a  fifth  harmonic  of  5 
per  cent  of  the  fundamental. 

In  Fig.  90  the  third  harmonic  is  in  phase,  in  Fig.  91  in  quadra- 
ture lagging,  and  in  Fig.  92  in  opposition  with  the  fundamental. 
A  gives  the  fundamental,  B  the  fundamental  with  the  thu'd  har- 
monic only,  and  C,  D,  E,  F,  the  waves  resulting  from  the  super- 
position of  the  fifth  harmonic  on  the  combination  of  funda- 
mental and  third  harmonic,  given  as  B.  In  C  the  fifth  harmonic 
is  in  opposition,  in  D  in  quadrature  lagging,  in  E  in  phase,  and 
in  F  in  quadrature  leading. 

We  see  here  round  tops  with  flat  zero  (Fig.  90C),  nearly 
triangular  waves  (Fig.  90£'),  approximate  half  circles  (Fig. 
92£'),  sine  waves  with  a  dent  at  the  top  (Fig.  92C),  and  vari- 
ous different  forms  of  saw  tooth. 

The  equations  of  these  waves  are: 

A:  y=100  cos /? 

B:  i/-100cos/?4-10cos3^9 

C:  ij=100  cos  ^  +  10  cos  3^3-5  cos  5^ 

D:  y^lOO  cos  yS  +  lO  cos  3/5-5  cos  (5/9  +  90  deg.) 

E:  7/  =100  cos  .9  +  10  cos  3..9  +  5  cos  5.9 


266  ENGINEERING  MATHEMATICS. 

A:  ?/ =100  COS/? 

B:  !/  =  100  cos  ^-10  cos  (3.? +  90  deg.) 

C:  2/ =100  cos  /?-10  cos  (3;? +  90  deg.) -5  cos  5/9 

D:  i/=100  cos  /9-10  cos  (3,5  +  90  deg.) -5  cos  (5/9  +  90  deg.) 

E:  ?/=100  cos  /9-10  cos  (3,5  +  90  deg.) +5  cos  5.5 

F:  ?/  =  100  cos  /?-10  cos  (3^  +  90  deg.) +5  cos  (5/9  +  90  deg.) 

A:  1/ =100  cos /9 

5.-  ?/=100  cos  .5-10  cos  3/9 

C;  7/ =100  cos  ^-10  cos  3.5-5  cos  5^ 

D:  y=im  cos  /9-10  cos  3^-5  cos  (5/9  +  90  deg.) 

E:  y=im  cos  /?-10  cos  3/9  +  5  cos  5/9 

Even  Harmonics. 

i6i.  Characteristic  of  the  wave-shape  distortion  of  even  har- 
monics is  that  the  wave  is  not  a  symmetrical  wave,  but  the 
two  half  waves  have  different  shapes,  and  the  characteristics 
of  the  negative  half  wave  are  opposite  to  those  of  the  positive. 
This  is  to  be  expected,  as  an  even  harmonic,  which  is  in  phase 
with  the  positive  half  wave  of  the  fundamental,  is  in  opposition 
with  the  negative;  when  leading  in  the  positive,  it  is  lagging 
in  the  negative,  and  inversely. 

Fig.  93  shows  the  effect  of  a  second  harmonic  of  30  per  cent 
of  the  fundamental  A,  superimposed  in  quadrature,  60  deg. 
phase  displacement,  30  deg.  displacement  and  in  phase,  in 
5,  C,  D  and  E  respectively. 

The  equations  of  these  waves  are : 

A:  ?/  =100  cos  ,9  and  y'  =30  cos  (2.5-90) 
B:  ?/=100  cos  /?  +  30  cos  (2/?-90) 
C:  ?/=100  cos  /9  +  30  cos  (2.5-60) 
D:  y  =100  cos  /9  +  30  cos  (2.5-30) 
E:  i/  =  100  cos  .5  +  30  cos  2.5 

Quadrature  combination  (Fig.  935)  gives  a  wave  where  the 
rising  side  is  flat,  the  decreasing  side  steep,  and  inversely  with 
the  other  half  wave.  C  and  D  give  a  peaked  wave  for  the  one,  a 
saw  tooth  for  the  other  half  wave,  and  E,  coincidence  of  phase 
of  fundamental  and  second  harmonic,  gives  a  combination  of 
one  peaked  half  wave  with  one  flat-top  or  double-peaked  wave. 


EMPIRICAL  CURVES. 


267 


Characteristic  of  C,  D  and  E  is,  that  the  two  half  waves  are 
of  unequal  length. 

In  general,  even  harmonics,  if  of  appreciable  value  are  easily 
recognized  by  the  difference  in  shape,  of  the  two  half  waves. 


30/^  Second  Harmonic 


Fig.  93.     Effect  of  Second  Harmonic. 


By  the  combination  of  the  second  harmonic  with  the  third 
harmonic  (or  the  fifth),  some  of  the  features  can  be  intensified, 
others  suppressed. 

An  illustration  hereof  is  shown  in  Fig,  94  in  the  production 


268 


ENGINEERING  MA  THEM  A  TICS. 


of  a  wave,  in  which  the  one  half  wave  is  a  short  high  peak,  the 
other  a  long  flat  top,  by  the  superposition  of  a  second  harmonic 
of  46.5  per  cent,  and  a  third  harmonic  of  10  per  cent  both  in 
phase  with  the  fundamental. 

A  gives  the  fundamental  sine  wave,  B  and  C  the  second  and 
third  harmonic,  D  the  combination  of  fundamental  and  second 


Second  &  Third  Harmonic 


Fig.  94.     Peak  and  Flat  Top  by  Second  and  Third  Harmonic. 

harmonic,  giving  a  double  peaked  negative  half  wave,  and  E  the 
addition  of  the  third  harmonic  to  the  wave  D.  Thereby  the 
double  peak  of  the  negative  half  wave  is  flatted  to  a  long  flat 
top,  and  the  peak  of  the  positive  half  wave  intensified  and 
shortened,  so  that  the  positive  maximum  is  about  two  and  one- 


EMPIRICAL  CURVES. 


269 


half  times  the  negative  maximum,  and  the  negative  half  wave 
nearly  75  per  cent  longer  than  the  positive  half  wave. 
The  equations  of  these  waves  are  given  by: 

A:  1/ =100  cos  ^ 

B:  ij  =46.5  cos  23 

C:  y  =10  cos  3/9 

D:  ij  =100  cos  -9  +  46.5  cos  2/9 

E:  ij  =100  cos  /9  +  46.5  cos  2^9  +  10  cos  3/9 

High  Harmonics. 

162.  Comparing  the  effect  of  the  fifth  harmonic,  Figs.  88  and 
89,  with  that  of  the  third  harmonic,  Figs.  85  and  86,  it  is  seen 


Fig.  95.     Effect  of  Seventh  Harmonic. 

that  a  fifth  harmonic,  even  if  very  small,  is  far  easier  distin- 
guished, that  is,  merges  less  into  the  fundamental  than  the  third 
harmonic.  Still  more  tliis  is  the  case  with  the  seventh  har- 
monic, as  shown  in  Fig.  95  in  phase  and  in  opposition,  of  10  per 
cent  intensity.  Tliis  is  to  be  expected:  sine  waves  which  do  not 
differ  very  much  in  frequency,  such  as  the  fundamental  and 
the  second  or  third  harmonic,  merge  into  each  other  and  form  a 
resultant  shape,  a  distorted  wave  of  characteristic  appearance, 


270  ENGINEERING  MATHEMATICS. 

while  sine  waA'es  of  ver}'  different  frequencies,  as  the  fundamen- 
tal and  its  eleventh  harmonic,  in  Fig.  96,  when  superimposed, 
remain  distinct  from  each  other;  the  general  shape  of  the  wave 
is  the  fundamental  sine,  and  the  high  harmonics  appear  as  rip- 
ples U])on  the  fundamental,  thus  giving  what  may  be  called  a 
corrugated  sine  wave.     By  counting  the  number  of  ripples  per 


Fig.  96.     Wave  in  which  Eleventh  Harmonic  Predominates. 

complete  wave,  or  per  half  wave,  the  order  of  the  harmonic 
can  then  rapidly  be  determined.  For  instance,  the  wave  shown 
in  Fig.  96  contains  mainly  the  eleventh  harmonic,  as  there  are 
eleven  ripples  per  wave.  The  wave  shown  by  the  oscillogram 
Fig.  97  shows  the  twenty-thirtl  harmonic,  etc. 


Fiu.  97.     C  D  23510.     Alternator  Wave  with  Single  High  Harmonic. 

\'ery  frequently  high  harmonics  appear  in  pairs  of  nearly  the 
same  frequency  and  intensity,  as  an  eleventh  and  a  thirteenth 
harmonic,  etc.  In  this  case,  the  ripples  in  the  wave  shape  show 
maxima,  where  the  two  harmonics  coincide,  and  nodes,  where 
the  two  harmonics  are  in  opposition.  The  })resence  of  nodes 
makes  the  counting  of  the  numl)er  of  ripples  per  complete  wave 
more  difficult.     A  convenient  method  of  procedure  in  this  case 


EMPIRICAL  CURVES.  271 

is,  to  measure  the  distance  or  space  between  the  maxima  of  one 
or  a  few  ripples  in  the  range  where  they  are  pronounced,  and 
count  the  number  of  nodes  per  cycle.  For  instance,  in  the 
wave.  Fig.  98,  the  space  of  two  ripples  is  about  60  deg.,  and  two 
nodes  exist  per  complete  wave.     60  deg.  for  two  ripples,  give 


Fig.  98.     Wave  in  which  Eleventh  and  Thirteenth  llarmonics  Predominate. 

2X =  12  ripples  per  complete  wave,  as  the  average  frequencv 

60 

of  the  two  existing  harmonics,  and  since  these  harmonics  differ 

by  2  (since  there  are  two  nodes),  their  order  is  the  eleventh  and 

the  thirteenth  harmonics. 

163.  This  method  of  determining  two  similar  harmonics,  with 


Fig.  99.     CD  23512.    Alternator   Wave  with   Two  Very  Unequal    Higli 

Harmonics. 

a  little  practice,  becomes  very  convenient  and  useful,  and  may 
frequently  be  used  visually  also,  in  determining  the  frequency 
of  hunting  of  synchronous  machines,  etc.  In  the  phenomenon 
of  hunting,  frecpiently  two  periods  are  superimj)osed,  a  forced 
frequency,  resulting  from  speed  of  generator,  etc.,  and  the 
natural  frequency  of  the  machine.  Counting  the  number  of 
impulses,  /,  per  minute,  and  the  number  of  nodes,  n,  gives  the 


272 


ENGINEERING  MATHEMATICS. 


two  frequencies:  /+-  and/—-;  and  as  one  of  these  frc(|uencie.s 

is  the  impressed  engine  frequency,  this  affords  a  check. 

Where  the  two  high  harmonics  of  nearly  equal  order,  as  the 
eleventh  and  the  thii-teenth  in  Fig.  98,  are  approximately  equal 
in  intensity,  at  the  nodes  the  ripples  practically  disappear, 
and  between  the  nodes  the  ripples  give  a  frequency  intermediate 
between  the  two  components:  Apparently  the  twelfth  harmonic 
in  Fig.  98.  In  this  case  the  two  constituents  are  easily  deter- 
mined: 12-1=11,  and  12  +  1=13. 

AVliere  of  the  two  constituents  one  is  greater  than  the  other 
the  wave  still  shows  nodes,  but  the  ripples  do  not  entirely  disap, 
pear  at  the  nodes,  but  merely  decrease,  that  is,  the  wave  show- 
a  sine  with  ripples  which  increase  and  decrease  along  the  waves 


Fig.  100.     CD  23511.     Alternator  Wave  with  Two    Nearly  Equal   High 

Harmonics. 

as  shown  by  the  oscillograms  99  and  100.  In  this  case,  one  of 
the  two  high  frecjuencies  is  given  by  counting  the  total  number 
of  ripples,  but  it  may  at  fii'st  be  in  doubt,  whether  the  other 
component  is  higher  or  lower  by  the  number  of  nodes.  The 
decision  then  is  made  by  considering  the  length  of  the  ripple  at 
the  node :  If  the  length  is  a  maximum  at  the  node,  the  secondary 
harmonic  is  of  higher  frequency  than  the  predominating  one; 
if  the  length  of  the  ripple  at  the  node  is  a  minimum,  the  second- 
ary frequency  is  lower  than  the  predominating  one.  This  is 
illustrated  in  Fig.  101.  In  this  figure,  A  and  B  represent  the 
tenth  and  twelfth  harmonic  of  a  wave,  respectively;  C  gives 
their  superposition  with  the  lower  harmonic  A  predominating, 
while  B  is  only  of  half  the  intensity  of  A.  D  gives  the  superposi- 
tion of  A  and  B  at  equal  intensity,  and  E  gives  the  super- 
position with  the  higher  frequency  B  predominating.  That  is, 
the  respective  equations  would  be: 


EMPIRICAL  CURVES.  273 

A:  y  =cos  10/? 

B:  y=cos  12^ 

C:  y=cos  10^9  +  0.5  cos  12^5 

D:  y=cos  10^  + cos  12-9 

E:  y  =0.5  cos  10/9  + cos  12-9 

As  seen,  in  C  the  half  wave  at  the  node  is  abnormally  long, 
showing  the  preponderance  of  the  lower  frequency,  in  E  abnor- 

Superposition  of  High  Harmonics 


Fig.  101.     Superposition  of  Two  High  Harmonics  of  Various  Intensities. 

mally  short,  showing  the  preponderance  of  the  higher  frequency. 
In  alternating-current  and  voltage  waves,  the  appearance  of 
two  successive  high  harmonics  is  quite  frequent.  For  instance, 
if  an  alternating  current  generator  contains  n  slots  per  pole, 
this  produces  in  the  voltage  wave  the  two  harmonics  of  orders 


274  ENGINEERING  MATHEMATICS. 

2n  — 1  and  2n-\-l.  Such  is  the  origm  of  the  harmonics  in  the 
oscillograms  Figs.  99  and  100. 

The  nature  of  the  increase  and  decrease  of  the  ripples  and  the 
formation  of  the  nodes  by  the  superposition  of  two  adjacent 
high  harmonics  is  best  seen  by  combining  their  expressions  trig- 
onometrically. 

Thus  the  harmonics: 

yi  =cos  (2n  — 1)/9 
and    2j2=cos  {2n  +  l)i3 

combined  give  the  resultant: 

=cos  (2m-1)/3  +  cos  (2n  +  l)/9 
=2  cos  /5  cos  2/1/9 

that  is,  give  a  wave  of  frequency  2?i  times  the  fundamental: 
cos  2n/9,  but  which  is  not  constant,  but  varies  in  intensity 
by  the  factor  2  cos  ^5. 

Not  infrequently  wave-shape  distortions  are  met,  which  are 
not  due  to  higher  harmonics  of  the  fundamental  wave,  but  are 
incommensurable  therewith.  In  this  case  there  are  two  entirely 
unrelated  frequencies.  This,  for  instance,  occurs  in  the  second- 
ary circuit  of  the  single-phase  induction  motor;  two  sets  of 
currents,  of  the  frequencies /^  and  (2/— /J  exist  (where /is  the 
primary  frequency  and/,  the  frequency  of  slip).  Of  this  nature, 
frequently,  is  the  distortion  produced  by  surges,  oscillations, 
arcing  grounds,  etc.,  in  electric  circuits;  it  is  a  combination  of 
the  natural  frequency  of  the  circuit  with  the  impressed  fre- 
quency. Telephonic  currents  commonly  show  such  multiple 
frequencies,  which  are  not  harmonics  of  each  other. 


CHAPTER  VII. 
NUMERICAL  CALCULATIONS. 

164.  Engineering  work  leads  to  more  or  less  extensive 
numerical  calculations,  when  applying  the  general  theoretical 
investigation  to  the  specific  cases  which  are  under  considera- 
tion.    Of  importance  in  such  engineering  calculations  are : 

(a)  The  method  of  calculation. 

(6)  The  degree  of  exactness  required  in  the  calculation. 

(c)  The  intelligibility  of  the  results. 

(d)  The  reliability  of  the  calculation. 

a.  Method  of  Calculation. 

Before  beginning  a  more  extensive  calculation,  it  is  desirable 
carefully  to  scrutinize  and  to  investigate  the  method,  to  find 
the  simplest  way,  since  frequently  by  a  suitable  method  and 
system  of  calculation  the  work  can  be  reduced  to  a  small  frac- 
tion of  what  it  would  otherwise  be,  and  what  appear  to  be 
hopelessly  complex  calculations  may  thus  be  carried  out 
quickly  and  expeditiously  by  a  proper  arrangement  of  the 
work.  Indeed,  the  most  important  part  of  engineering  work — and 
also  of  other  scientific  work — is  the  determination  of  the  method 
of  attacking  the  problem,  whatever  it  may  be,  whether  an 
experimental  investigation,  or  a  theoretical  calculation.  It  is 
very  rarely  that  important  problems  can  be  solved  by  a  direct 
attack,  by  brutally  forcing  a  solution,  and  then  only  by  wasting 
a  large  amount  of  work  unnecessarily.  It  is  by  the  choice  of 
a  suitable  method  of  attack,  that  intricate  problems  are  reduced 
to  simple  phenomena,  and  then  easily  solved;  frequently  in 
such  cases  requiring  no  solution  at  all,  but  being  obvious  when 
looked  at  from  the  proper  viewpoint. 

Before  attacking  a  more  complicated  problem  experimentally 
or  theoretically,  considerable  time  and  study  should  thus  first  be 
devoted  to  the  determination  of  a  suitable  method  of  attack. 

275 


276  ENGINEERING  MATHEMATICS. 

The  next  then,  in  cases  where  considerable  numerical  calcu- 
lations are  required,  is  the  method  of  calculation.  The  most 
convenient  one  usually  is  the  arrangement  in  tabular  form. 

As  example,  consider  the  problem  of  calculating  the  regula- 
tion of  a  60,000- volt  transmission  line,  of  r  =  60  ohms  resist- 
ance, a:  =  135  ohms  inductive  reactance,  and  6  =  0.0012  conden- 
sive  susceptance,  for  various  values  of  non-inductive,  inductive, 
and  condensive  load. 

Starting  with  the  complete  equations  of  the  long-distance 
transmission  line,  as  given  in  "  Theory  and  Calculation  of 
Transient  Electric  Phenomena  and  Oscillations,"  Section  III, 
paragraph  9,  and  considering  that  for  every  one  of  the  various 
power-factors,  lag,  and  lead,  a  sufficient  number  of  values 
have  to  be  calculated  to  give  a  curve,  the  amount  of  work 
appears  hopelessly  large. 

However,  without  loss  of  engineering  exactness,  the  equa- 
tion of  the  transmission  line  can  be  simplified  by  approxima- 
tion, as  discussed  in  Chapter  V,  paragraph  123,  to  the  form, 


^i  =  j^o|l+^|+  ZjJl  + 


ZY 

0 


h  =  Io  {!+-[] +YeJi+^\, 


(1) 


where  Eq,  Iq  are  voltage  and  current,  respectively  at  the  step- 
down  end,  El,  /i  at  the  ^tep-up  end  of  the  line;  and 

Z  =  r-\-jx  =  00 -{-135]  is  the  total  line  impedance; 

Y  =  g-\-jb=  +0.0012/  is  the  total  shunted  line  admittance. 

Herefrom  follow  the  numerical  values : 

,     ZY     ,     (60  +  135/)(+0.0012y) 
1  +"2-  =  1  +^ -^ 

=  ]  +  0.036/-  0.0«1  =  0.9194  0.036/; 

7Y 
1+^  =  1  +  0.012/-  0.027  =  0.973  ^  0.012/; 


NUMERICAL  CALCULATIONS. 


277 


Z\ 


ZY] 
(i 


1  +^p-    =(60  +  135j)(0.973+0.012?) 


=  58.4+vl.72j-f-131.1j- 1.62  =  56.8+131.8?; 


7(l+:^^|=(+0.0012y)(0.973+0.012j) 
I  t)   J       ^ 

=  + 0.001168/- 0.0000144  =  (- 0.0144 +  1.168/)10-3 

hence,   substituting  in    (1),   the   following  equations   may   be 
written : 

^1  =  (0.919  '  0.036j)£'o+(56.8  +  131.8y)/o  =  ^+B;  1 

/i  =  (0.919+0.036y)/o  -  (0.0144  -1.168/)£;ol0-3  =  C-Z).  /  ^^^ 
165-   Now  the   work  of   calculating   a  series   of  numerical 
values  is  continued  in  tabular  form,  as  follows: 

1.  100  PER  CENT  Power-factor. 

£0=60  kv.  at  step-down  end  of  line. 

A  =  (0.919 +0.036;)Bo=  55.1  +  2,2;  kv. 

D=  (0.0144- 1.168/)'£o  10-3  =  0.9-70.1/  amp. 


/(,  amp. 

B  kv. 

=  A+B. 

f,2  +  e22  =  e2. 

e 

ei 

—  =  tane. 

ei 

^e. 

0 

0 

55.1+   2.2; 

3036+      5  =  3041 

55.1 

+  0.040 

+  2.3 

20 

1.1+   2.6; 

56.2+  4.8; 

3158+    23  =  3181 

56.4 

+  0.085 

+  4.9 

40 

2.3+    5.3; 

57.4+   9.5; 

3295+    56  =  3351 

57.9 

+  0.131 

+  7.5 

60 

3.4+    7.9; 

58.5  +  10.1; 

3422  +  102  =  3524 

59.4 

+  0.173 

+  9.9 

80 

4.5+10.5; 

59.6  +  12.7; 

3552  +  161  =  3713 

60.9 

+  0.213 

+12.0 

100 

5.7+13.2; 

60.8  +  15.4; 

3697  +  237  =  3934 

62.7 

+  0.253 

+14.2 

120 

6.8+15.8; 

61.9  +  18.0; 

3832  +  324=4156 

64.5 

+  0.291 

+16.3 

h 
amp. 

C  amp. 

Ji  =  ii+;i2 
=  C-D 

ri2  +  t22  =  l2 

i 

i2 

-  =  tani 
ii 

Ai 

2ii- 
+  88.6 

Power- 
factor 

0.024 

0 

0 

-0.7-90.1; 

4914+1          =   4915 

70.1 

-78 

-89.1 

= 

+  90.9 

20 

18.4+0.7; 

17.5+70.8; 

5013+   306=    5319 

72.9 

+4.04 

+76.3 

+71.4 

0.332 

40 

36.8  +  1.4; 

35.9+71.5; 

5112  +  1289=    6401 

80.0 

+  1.99 

+83.4 

+  55.9 

0  558 

60 

55.1  +  2.2; 

54.2+72.3; 

5227  +  2938=    8165 

90.4 

+  1.33 

+53.1 

+  43.2 

0  728 

80 

73.5  +  2.9; 

72.6+73.0; 

5329  +  5271=10600  103.0 

+  1.055 

+45.2 

+  33.2 

0  837 

100 

91.9  +  3.6; 

91.0+73.9; 

8281  +  5432=13713 

117.1 

+  0.811 

4  39.1 

+  24.9 

0  907 

120 

llO.S  +  4.3; 

109.4+74.4; 

11969  +  5535=17504 

132.3 

+0.680 

J  34.1 

+  17.8 

0  952 
lead 

278 


ENGINEERING  MATHEMATICS. 


ei  =  60  kv.  at  step-up  end  of  line. 

Red.  Factor, 

/o 
amp. 

e 
60 

amp. 

kv. 

amp. 

Power-Factor. 

0 

0.918 

0 

65.5 

76.4 

0.024 

20 

0.940 

21.3 

63.8 

77.5 

0.332 

40 

0.965 

41.4 

62.1 

82.9 

0.558 

60 

0.990 

60.6 

60.6 

91.4 

0.728 

80 

1.015 

78.8 

59.1 

101.5 

0.837 

100 

1.045 

95  7 

57.5 

112.3 

0.907 

120 

1.075 

111.7 

55  8 

122.8 

0.952 
lead 

Curv 

68  of  Iq,  fQ,  i,,  COS  e,  plotted  in  Fig.  86. 

. 

2.  90  Per  Cent  Power-Factor,  Lag. 


cos  ^  =  0.9;    sin(9  =  Vl-0.92  =  0.436; 

7o  =  io(cos  e-j^\n  ^)=2o(0.9 -0.436/); 

El  =  (0.919+  0.036j>o  +  (56.8  +  131.8/)  (0.9  -0.436/H'o 

=  (0.919+ 0.036/)eo  + (108.5 +  93.8/Ho  =  ^  +5'; 
h  =  (0.919+  0.036/)  (0.9  -0.436/)io-  (0.0144  -  1.168/)eolO-3 
'  =  (0.843 -0.366/)io- (0.0144 -1.168/)eolO-3  =  C'-D, 

and  now  the  table  is  calculated  in  the  same  manner  as  under  1. 
Then   corresponding   tables   are    calculated,    in   the   same 
manner,  for  power-factor,    =0.8  and   =0.7,  respectively,   lag, 
and  for  power-factor  =0.9,  0.8,  0.7,  lead;  that  is,  for 


cos  ^  +  / sin  ^  =  0.8-0.6/; 
0.7-0.714/; 
0.9+0.436/; 
0.8+0.6/; 
0.7+0.714/. 

Then  curves  are  plotted  for  all  seven  values  of  power-factor, 
from  0.7  lag  to  0.7  lead. 

From  these  curves,  for  a  number  of  values  of  to,  for  instance, 
io  =  20,  40,  60,  80,  100,  numerical  values  of  ii,  eo,  cos  6,  are 


numekiCal  calculations. 


279 


taken,  and  plotted  as  curves,  which,  for  the  same  voltage 
ei  =  60  at  the  step-up  end,  give  ii,  eo,  and  cos  d,  for  the  same 
value  io,  that  is,  give  the  regulation  of  the  line  at  constant 
current  output  for  varying  power-factor. 


b.  Accuracy  of  Calculation. 

166.  Not  all  engineering  calculations  require  the  same 
degree  of  accuracy.  AVhen  calculating  the  efficiency  of  a  large 
alternator  it  may  be  of  importance  to  determine  whether  it  is 
97.7  or  97.8  per  cent,  that  is,  an  accuracy  within  one-tenth 
per  cent  may  be  required;  in  other  cases,  as  for  instance, 
when  estimating  the  voltage  which  may  be  produced  in  an 
electric  circuit  by  a  line  disturbance,  it  may  be  sufficient  to 


1001.00 


80  0.80 


0.60 


0.10 


0.30 


Fig.  102.     Transmission  Line  Characteristics. 

determine  whether  this  voltage  would  be  limited  to  double 
the  normal  circuit  voltage,  or  whether  it  might  be  5  or  10 
times  the  normal  voltage. 

In  general,  according  to  the  degree  of  accuracy,  engineering 
calculations  may  be  roughly  divided  into  three  classes : 


280  ENGINEERING  MATHEMATICS. 

(a)  Estimation  of  the  magnitude  of  an  effect;  that  is, 
determining  approximate  numerical  values  within  25,  50,  or 
100  per  cent.  Very  frequently  such  very  rough  approximation 
is  sufficient,  and  is  all  that  can  be  expected  or  calculated. 
For  instance,  when  investigating  the  short-circuit  current  of  an 
electric  generating  system,  it  is  of  importance  to  know  whether 
this  current  is  3  or  4  times  normal  current,  or  whether  it  is 
40  to  50  times  normal  current,  but  it  is  immaterial  w^hether 
it  is  45  to  46  or  50  times  normal.  In  studying  lightning 
phenomena,  and,  in  general,  abnormal  voltages  in  electric 
systems,  calculating  the  discharge  capacity  of  lightning  arres- 
ters, etc.,  the  magnitude  of  the  quantity  is  often  sufficient.  In 
calculating  the  critical  speed  of  turbine  alternators,  or  the 
natural  period  of  oscillation  of  synchronous  machines,  the 
same  applies,  since  it  is  of  importance  only  to  see  that  these 
speeds  are  sufficiently  remote  from  the  normal  operating  speed 
to  give  no  trouble  in  operation. 

(6)  Approximate  calculation,  requiring  an  accuracy  of  one 
or  a  few  per  cent  only;  a  large  part  of  engineering  calcu- 
lations fall  in  this  class,  especially  calculations  in  the  realm  of 
design.  Although,  frequently,  a  higher  accuracy  could  be 
reached  in  the  calculation  proper,  it  would  be  of  no  value, 
since  the  data  on  which  the  calculations  are  based  are  sus- 
ceptible to  variations  beyond  control,  due  to  variation  in  the 
material,  in  the  mechanical  dimensions,  etc. 

Thus,  for  instance,  the  exciting  current  of  induction  motors 
may  vary  by  several  per  cent,  due  to  variations  of  the  length 
of  air  gap,  so  small  as  to  be  beyond  the  limits  of  constructive 
accuracy,  and  a  calculation  exact  to  a  fraction  of  one  per  cent, 
while  theoretically  possible,  thus  w^ould  be  practically  useless. 
The  calculation  of  the  ampere-turns  required  for  the  shunt 
field  excitation,  or  for  the  series  field  of  a  direct-current 
generator  needs  only  moderate  exactness,  as  variations  in  the 
magnetic  material,  in  the  speed  regulation  of  the  driving 
power,  etc.,  produce  differences  amounting  to  several  per 
cent. 

(c)  Exact  engineering  calculations,  as,  for  instance,  the 
calculations  of  the  efficiency  of  apparatus,  the  regulation  of 
transformers,  the  characteristic  curves  of  induction  motors, 
etc.  These  are  determined  with  an  accuracy  frequently  amount- 
ing to  one-tenth  of  one  per  cent  and  even  greater. 


NUMERICAL  CALCULATIONS.  281 

Even  for  most  exact  engineering  calculaf.ons,  the  accuracy 
of  the  slide  rule  is  usualW  sufficient,  if  intelligently  used,  that 
is,  used  so  as  to  get  the  greatest  accuracy.  For  accurate  calcu- 
lations, preferably  the  glass  slide  should  not  be  used,  but  the 
result  interpolated  by  the  eye. 

Thereby  an  accuracy  within  }  per  cent  can  easily  be  main- 
tained. 

For  most  engineering  calculations,  logarithmic  tables  are 
sufficient  for  three  decimals,  if  intelhgently  used,  and  as  such 
tables  can  be  contained  on  a  single  page,  their  use  makes  the 
calculation  very  much  more  expeditious  than  tables  of  more 
decimals.  The  same  apphes  to  trigonometric  tables:  tables 
of  the  trigonometric  functions  (not  their  logarithms)  of  three 
decimals  I  find  most  convenient  for  most  cases,  given  from 
degree  to  degree,  and  using  decimal  fractions  of  the  tlegrees 
(not  minutes  and  seconds).* 

Expedition  in  engineering  calculations  thus  requires  the  use 
of  tools  of  no  higher  accuracy  than  required  in  the  result,  and 
such  are  the  slide  rules,  and  the  three  decimal  logarithmic  ami 
trigonometric  tables.  The  use  of  these,  however,  make  it 
neccessary  to  guard  in  the  calculation  against  a  loss  of  accuracy. 

Such  loss  of  accuracy  occurs  in  subtracting  or  dividing  two 
terms  which  are  nearly  ecjual,  in  some  logarithmic  operations, 
solution  of  equation,  etc,,  and  in  such  cases  either  a  higher 
accuracy  of  calculation  must  be  employed — seven  decimal 
logarithmic  tables,  etc. — or  the  operation,  which  lowers  the 
accuracy,  avoided.  The  latter  can  usually  be  done.  For 
instance,  in  dividing  297  by  283  by  the  slide  rule,  the  proper 
way  is  to  divide  297-283  =  14  by  283,  and  add  the  result 
to  1. 

It  is  in  the  methods  of  calculation  that  experience  and  judg- 
ment and  skill  in  efficiency  of  arrangement  of  numerical  calcu- 
lations is  most  marked. 

167.  While  the  calculations  are  unsatisfactory,  if  not  carried 
out  with  the  degree  of  exactness  which  is  feasible  and  desirable, 
it  is  equally  wrong  to  give  numerical  values  with  a  number  of 

*  This  obviously  does  not  apply  to  some  classes  of  engineering  work,  in 
which  a  much  higher  accuracy  of  trigonometric  functions  is  required,  aa 
trigonometric  surveying,  etc. 


282  ENGINEERING  MATHEMATICS. 

ciphers  greater  than  the  method  or  the  purpose  of  the  calcula- 
tion warrants.  For  instance,  if  in  the  design  of  a  direct-current 
generator,  the  calculated  field  ampere-turns  are  given  as  9738, 
such  a  numerical  value  destroys  the  confidence  in  the  work  of 
the  calculator  or  designer,  as  it  implies  an  accuracy  greater 
than  possible,  and  thereby  shows  a  lack  of  judgment. 

The  number  of  ciphers  in  which  the  result  of  calculation  is 
given  should  signify  the  exactness,  In  this  respect  two 
systems  are  in  use: 

(a)  Numerical  values  are  given  wdth  one  more  decimal 
than  warranted  by  the  probable  error  of  the  result;  that  is, 
the  decimal  before  the  last  is  correct,  but  the  last  decimal  may 
be  wrong  by  several  units.  This  method  is  usually  employed 
in  astronomy,  physics,  etc. 

(6)  Numerical  values  are  given  with  as  many  decimals  as 
the  accuracy  of  the  calculation  warrants;  that  is,  the  last 
decimal  is  probably  correct  within  half  a  unit.  For  instance, 
an  efficiency  of  86  per  cent  means  an  efficiency  between  85.5 
and  86.5  per  cent;  an  efficiency  of  97.3  per  cent  means  an 
efficiency  between  97.25  and  97.35  per  cent,  etc.  This  system 
is  generally  used  in  engineering  calculations.  To  get  accuracy 
of  the  last  decimal  of  the  result,  the  calculations  then  must 
be  carried  out  for  one  more  decimal  than  given  in  the  result. 
For  instance,  when  calculating  the  efficiency  by  adding  the 
various  percentages  of  losses,  data  like  the  following  may  be 
given : 

Core  loss 2.73  per  cent 

r^r 1.06       " 

Friction 0.93 

Total 4.72 

Efficiency 100-4.72  =  95.38 

Approximately 95.4  " 

It  is  obvious  that  throughout  the    same  calculation  the 
same  degree  of  accuracy  must  be  observed. 
It  follows  herefrom  that  the  values : 

2h;    2.5;     2.50;     2.500, 


NUMERICAL  CALCULATIONS.  283 

while  mathematically  equal,  are  not  equal  in  their  meaning  as 
an  engineering  result : 

2.5      means  between  2.45    and  2.55; 
2.50    means  between  2.495    and  2.505; 
2.500  means  between  2.4995  and  2.5005; 

while  2^  gives  no  clue  to  the  accuracy  of  the  value. 

Thus  it  is  not  permissible  to  add  zeros,  or  drop  zeros  at 
the  end  of  numerical  values,  nor  is  it  permissible,  for  instance, 
to  replace  fractions  as  1/16  by  0.0625,  without  changing  the 
meaning  of  the  numerical  value,  as  regards  its  accuracy. 
This  is  not  always  realized,  and  especially  in  the  reduction  of 
common  fractions  to  decimals  an  unjustified  laxness  exists 
which  impairs  the  reliability  of  the  results.  For  instance,  if 
in  an  arc  lamp  the  arc  length,  for  which  the  mechanism  is 
adjusted,  is  stated  to  be  0.8125  inch,  such  a  statement  is 
ridiculous,  as  no  arc  lamp  mechanism  can  control  for  one-tenth 
as  great  an  accuracy  as  implied  in  this  numerical  value:  the 
value  is  an  unjustified  translation  from  13/16  inch. 

The  principle  thus  should  be  adhered  to,  that  all  calcula- 
tions are  carried  out  for  one  decimal  more  than  the  exactness 
required  or  feasible,  and  in  the  result  the  last  decimal  dropped: 
that  is,  the  result  given  so  that  the  last  decimal  is  probably 
correct  within  half  a  unit. 

c.  Intelligibility  of  Engineering  Data  and  Engineering  Reports. 

1 68.  In  engineering  calculations  the  value  of  the  results 
mainly  depends  on  the  information  derived  from  them,  that  is, 
on  their  intelligibility.  To  make  the  numerical  results  and 
their  meaning  as  intelligible  as  possible,  it  thus  is  desirable, 
whenever  a  series  of  values  are  calculated,  to  carefully  arrange 
them  in  tables  and  plot  them  in  a  curve  or  in  curves.  The 
latter  is  necessary,  since  for  most  engineers  the  plotted  curve 
gives  a  much  better  conception  of  the  shape  and  the  variation 
of  a  quantity  than  numerical  tables. 

Even  where  only  a  single  point  is  required,  as  the  core 
loss  at  full  load,  or  the  excitation  of  an  electric  generator  at 
rated  voltage,  it  is  generally  preferable  to  calculate  a  few 


284 


ENGINEERING  MA  THEM  A  TICS. 


Volts 

— 





-500 

0 

2 

0 

4 

0 

fi 

n 

8 

1 

0 

Fig.  103.     Compounding  Curve. 


points  near  the  desired  value,  so  as  to  get  at  least  a  short  piece 
of  curve  including  the  desired  point. 

The  main  advantage,  and  foremost  purpose  of  curve  plotting 
thus  is  to  show  the  shape  of  the  function,  and  thereby  give 
a  clearer  conception  of  it  ; 
but  for  recording  numerical 
values,  and  deriving  numer- 
ical values  from  it,  the  plotted 
curve  is  inferior  to  the  table, 
due  to  the  limited  accuracy 
po.^sible  in  a  plotted  curve, 
and  the  further  inaccuracy 
resulting  when  drawing  a 
curve  through  the  plotted  cal- 
culated points.  To  some 
extent,  the  numerical  values 
as  taken  from  a  plotted  curve, 
depend  on  the  particular 
kind  of  curve  rule  used  in 
plotting  the  curve. 

In  general,  curves  are  used  for  two  different  purposes,  and 
on  the  purpose  for  which  the  curve  is  plotted,  should  depend 
the  method  of  plotting,  as  the  scale,  the  zero  values,  etc. 

^^'hen  curves  are  used  to 
illustrate  the  shape  of  the 
function,  so  as  to  show  how 
much  and  in  what  manner  a 
quantity  varies  as  function 
of  another,  large  divisions  of 
inconspicuous  cross-section- 
ing are  desirable,  but  it  is 
essential  that  the  cross- 
sectioning  should  extend  to 
the  zero  values  of  the  func- 
tion, even  if  the  numerical 
values  do  not  extend  so 
far,  since  otherwise  a  wrong 
impression  would  be  con- 
ferred. As  illustrations  are  plotted  in  Figs.  103  and  104,  the 
compounding  curve  of  a  direct-current  generator.    The  arrange- 


V 

olts 
-550 

^ 

^ 

^ 

^ 

y 

y 

/■ 

0 

2 

0 

4 

0 

6 

0 

8 

1 

0 

Fig.  104.     Compounding  Curve. 


NUMERICAL  CALCULATIONS. 


285 


ment  in  Fig.  103  is  correct;  it  shows  the  relative  variation 
of  voltage  as  function  of  the  load.  Fig.  104,  in  which  the 
cross-sectioning  does  not  be^in  at  the  scale  zero,  confers  the 


._ 

^^ 

^ 

/ 

/ 

1 

■h 

f 

1 

C7- 

1 

3 

2 

) 

3 

0      1      41 

) 

5 

^ 

el) 

7 

0 

Fig.  105.     Curve  Plotted  to  show  Characteristic  Shape. 


Fig.  106.     Curve  Plotted  for  Use  as  Design  Data. 

wrong  impression  that  the  variation  of  voltage  is  far  greater 
than  it  really  is. 

When   curves    are   used    to   record    numerical  values   and 
derive  them  from  the  curve,  as,  for  instance,  is  commonly  the 


^86 


ENGINEERING  MATHEMATICS. 


case  with  magnetization  curves,  it  is  unnecessary  to  have  the  zero 
of  the  function  coincide  with  the  zero  of  the  cross-sectioning,  but 
rather  preferable  not  to  have  it  so,  if  thereby  a  better  scale  of 
the  curve  can  be  secured.  It  is  desirable,  however,  to  use  suffi- 
ciently small  cross-sectioning  to  make  it  possible  to  take  numer- 
ical values  from  the  curve  with  good  accuracy.  This  is  illus- 
trated by  Figs.  105  and  106.  Both  show  the  magnetic  charac- 
teristic of  soft  steel,  for  the  range  above B  =  8000,  in  which  it  is 
usually  employed.  Fig.  105  shows  the  proper  way  of  plotting  for 
showing  the  shape  of  the  function,  Fig.  106  the  proper  way  of 
plotting  for  use  of  the  curve  to  derive  numerical  values  therefrom. 


^ 

V 

\ 

\ 

\ 

— N 

V 

\ 

\ 

\ 

I 

\ 

\ 

II 

\ 

\, 

N 

n\ 

N 

K 

V 

s 

^^^ 

\ 

^ 

:^ 

^. 

m 

^ 

--^ 

"^ 

■- 

■ — ■ 

— 





- 

Fig.  107.     Same  Function  Plotted  to  Different  Scales;  I  is  correct. 

169.  Curves  should  be  plotted  in  such  a  manner  as  to  show 
the  quantity  which  they  represent,  and  its  variation,  as  well  as 
possible.     Two  features  are  desirable  herefor: 

1.  To  use  such  a  scale  that  the  average  slope  of  the  curve, 
or  at  least  of  the  more  important  part  of  it,  does  not  differ 
much  from  45  deg.  Hereby  variations  of  curvature  are  best 
shown.  To  illustrate  this,  the  exponential  function  y  =  c~''  is 
plotted  in  three  different  scales,  as  curves  I,  II,  III,  in  Fig.  107. 
Curve  I  has  the  proper  scale. 

2.  To  use  such  a  scale,  that  the  total  range  of  ordinates  is 
not  much  different  from  the  total  range  of  abscissas.  Thus 
when  plotting  the  power-factor  of  an  induction  motor,  in 
Fig.  108,  curve  I  is  preferable  to  curves  II  or  III. 


NUMERICAL  CALCULATIONS. 


287 


These  two  requirements  frequently  are  at  variance  with 
each  other,  and  then  a  compromise  has  to  be  made  between 
them,  that  is,  such  a  scale  chosen  that  the  total  ranges  of  the 
two  coordinates  do  not  differ  much,  and  at  the  same  time 
the  average  slope  of  the  curve  is  not  far  from  45  deg.  This 
usually  leads  to  a  somewhat  rectangular  area  covered  by  the 
curve,  as  shown,  for  instance,  by  curve  I,  in  Fig.  107. 

It  is  obvious  that,  where  the  inherent  nature  of  the  curve 
is  incompatible  with  45  degree  slope,  this  rule  does  not  apply. 
Such  for  instance  is  the  case  with  instrument  cahbration  curves, 
which  inherently  are  essentially  horizontal  lines,  with  curves 
like  the  slip  of  induction  motors,  etc. 

As  regards  to  the  magnitude  of  the  scale  of  plotting,  the  larger 
the  scale,  the  plainer  obviously  is  the  curve.  It  must  be  kept  in 
mind,  however,  that  it  would  be  wrong  to  use  a  scale  which  is 
materially    larger    than    the   accuracy  of  the  values   plotted. 

Thus  for  instance,  in 
plotting  the  calibration 
curve  of  an  instrument, 
if  the  accuracy  of  the 
calibration  is  not  greater 
than  .05  per  cent,  it 
would  be  wrong  to  use 
.01  per  cent  as  the  unit 
of  ordinate  scale. 

In  curve  plotting,  a 
scale  should  be  used 
which  is  easily  read. 
Hence,  only  full  scale, 
double  scale,  and  half 
scale  should  be  used. 
Triple  scale  and  one- 
third  scale  are  practi- 
cally unreadable,  and  should  therefore  never  be  used.  Quadruple 
scale  and  quarter  scale  are  difficult  to  read  and  therefore  unde- 
sirable, and  are  generally  unnecessary,  since  quadruple  scale  is 
not  much  different  from  half  scale  with  a  ten  times  smaller  unit, 
and  quarter  scale  not  much  different  from  double  scale  of  a  ten 
times  larger  unit. 

170.  In  plotting  a  curve  to  show  a  relation  y=f{x),  in  gen- 
eral X  and  y  should  be  plotted  directly,  on  ordinary  coordinate 


7 

y 

"^ 

T 

^ 

/ 

/ 

/ 

y' 

/ 

/ 

/ 

i 

' , 

/ 

/ 

/ 

III 

— 

^ 

// 

f 

^ 

"^ 

u 

/ 

^ 

h 

r 

Fig.  108.     Same  Function  Plotted   to  Dif- 
ferent Scales;  I  is  Correct. 


288  ENGINEERING  MATHEMATICS. 

paper,  but  not  log  x,  or  y-,  or  logarithmic  paper  used,  etc.,  as 
this  would  not  show  the  shape  of  the  relation  y=f(x).  Using 
for  instance  semi-logarithmic  paper,  that  is,  with  logarith- 
mic abscissae  and  ordinary  ordinates,  the  plotted  curve  would 
show  the  shape  of  the  relation  y=f  (log  x),  etc.  The  use  of 
logarithmic  paper,  or  the  use  of  y-,  or  -y/x  as  coordinate,  etc.,  is 
justified  only  where  the  purpose  is  to  show  the  relation  between 
y  and  log  x,  or  between  y-  and  x,  or  between  y  and  -y/x,  etc.,  as  is 
the  case  when  investigating  the  equation  of  an  empir  cal  curve, 
or  when  intending  to  show  some  particular  feature  of  the  relation 
y=f{x).  Thus  for  instance  when  plotting  the  power  p  consumed 
by  corona  in  a  high  potential  transmission  line,  as  function  of  the 
line  voltage  e,  by  using  -y/p  as  ordinate,  a  straight  line  results. 
Also  where  some  particular  function  of  one  of  the  coordinates, 
as  log  X,  gives  a  more  rational  relation,  it  ma}"  be  used  instead 
of  X.  Thus  for  instance  in  radiation  curves,  or  when  expressing 
velocity  as  function  of  wave  length  or  frequency,  or  expressing 
attenuation  of  a  wireless  wave,  etc.,  the  log  of  wave  length  or 
frequency,  that  is,  the  geometric  scale  (as  used  in  the  theory  of 
sound,  with  the  octave  as  unit)  is  more  rational  and  therefore 
preferable. 

Sometimes  the  values  of  a  relationship  extend  over  such  a 
wide  range  as  to  make  it  impossible  to  represent  all  of  them  in 
one  curve,  and  then  a  number  of  curves  may  have  to  be  used, 
with  different  scales.  In  such  cases,  the  logarithmic  scale  often 
brings  all  values  within  one  curve  without  improperl}'  crowding, 
and  especially  where  the  purpose  of  curve  plotting  is  not  so 
much  to  show  the  shape  of  the  relation,  as  to  record  for  the  pur- 
pose of  taking  numerical  values  from  the  curve,  the  latter  ar- 
rangement, that  is,  the  use  of  logarithmic  or  semi-logarithmic 
paper  may  be  desirable.  Thus  the  magnetic  characteristic  of 
iron  is  used  over  a  range  of  field  intensities  from  very  few  am- 
pere turns  per  cm.  in  transformers,  to  thousands  of  ampere 
turns,  in  tooth  densities  of  railway  motors,  and  the  magnetic 
characteristic  thus  is  either  represented  by  three  curves  with 
different  scales,  of  ratios  1-^10-^100,  as  shown  in  Fig.  109,  or 
the  log  of  field  intensity  used  as  abscissae,  that  is,  semi-logarith- 
mic paper,  with  logarithmic  scale  as  abscissae,  and  regular  scale 
as  ordinates,  as  shown  in  Fig.  110. 

It  must  be  realized  that  the  logarithmic  or  geometrical  scale 


NUMERICAL  CALCULATIONS. 


289 


Fig.  109. 


B 

-23000 

1 

2200(L 

Maf 

n 

'tic 

Chare 

c 

c 

ristic 

of 

^ 

^ 

^' 

21000- 

1   Si^lijcon  St 
(Semilogarithln 

e 
c 

1 

Scale) 

** 

»• 

- 

20000 

x^ 

.^ 

19000. 

^ 

x' 

18000- 

/ 

/ 

13000. 

/ 

/ 

16000- 

^ 

^ 

^ 

15000- 

,-- 

^ 

1 

, 

I«00- 
13000 

^^ 

•^ 

^ 

^ 

^ 

12003 

^ 

^ 

- 

UOOO 

/ 

Z' 

JOOOO- 
900qZ 

/ 

/ 

MXfi 

f 

i 

S 

1 

)  1 

2 

1 

G 

a 

0        3 

)     4 

)    5 

D6 

0 

8 

) 

1 

X)^ 

^ 

g 

\     %   i 

?  §^ 

\ 

i 

?i 

Fig.  110. 


290  ENGINEERING  MATHEMATICS. 

— in  which  equal  divisions  represent  not  equal  values  of  the 
quantity,  but  equal  fractions  of  the  quantity — is  somewhat 
less  easy  to  read  than  common  scale.  However,  as  it  is  the  same 
scale  as  the  slide  rule,  this  is  not  a  serious  objection. 

A  disadvantage  of  the  logarithmic  scale  is  that  it  cannot 
extend  down  to  zero,  and  relations  in  which  the  entire  range 
down  to  zero  requires  consideration,  thus  are  not  well  suited 
for  the  use  of  logarithmic  scale. 

171.  Any  engineering  calculation  on  which  it  is  worth 
while  to  devote  any  time,  is  worth  being  recorded  with  suffi- 
cient completeness  to  be  generally  intelligible.  Very  often  in 
making  calculations  the  data  on  which  the  calculation  is  based, 
the  subject  and  the  purpose  of  the  calculation  are  given  incom- 
pletely or  not  at  all,  since  they  are  familiar  to  the  calculator  at 
the  time  of  calculation.  The  calculation  thus  would  be  unin- 
telligible to  any  other  engineer,  and  usually  becomes  unintelli- 
gible even  to  the  calculator  in  a  few  weeks. 

In  addition  to  the  name  and  the  date,  all  calculations  should 
be  accompanied  by  a  complete  record  of  the  object  and  purpose 
of  the  calculation,  the  apparatus,  the  assumptions  made,  the 
data  used,  reference  to  other  calculations  or  data  employed, 
etc.,  in  short,  they  should  include  all  the  information  required 
to  make  the  calculation  intelligible  to  another  engineer  without 
further  information  besides  that  contained  in  the  calculations, 
or  in  the  references  given  therein.  The  small  amount  of  time 
and  work  required  to  do  this  is  negligible  compared  with  the 
increased  utility  of  the  calculation. 

Tables  and  curves  belonging  to  the  calculation  should  in 
the  same  way  be  completely  identified  with  it  and  contain 
sufficient  data  to  be  intelligible. 

171A.  Engineering  investigations  evidently  are  of  no  value, 
unless  they  can  be  communicated  to  those  to  whom  they  are  of 
interest.  Thus  the  engineering  report  is  an  essential  and  im- 
portant part  of  the  work.  If  therefore  occasionally  an  engineer 
or  scientist  is  met,  who  is  so  much  interested  in  the  investigating 
work,  that  he  hates  to  "waste"  the  time  of  making  proper  and 
complete  reports,  this  is  a  very  foolish  attitude,  since  in  general 
it  destroys  the  value  of  the  work. 

As  practically  every  engineering  investigation  is  of  interest 
and  importance  to  different  classes  of  people,  as  a  rule  not  one, 


NUMERICAL  CALCULATIONS.  291 

but  several  reports  must  be  written  to  make  the  most  use  of 
the  work;  the  scientific  record  of  the  research  would  be  of  no 
more  value  to  the  financial  interests  considering  the  industrial 
development  of  the  work  than  the  report  to  the  financial  or 
administrative  body  would  b^  of  value  to  the  scientist,  who 
considers  repeating  and  continuing  the  investigation. 

In  general  thus  three  classes  of  engineering  reports  can  be 
distinguished,  and  all  three  reports  should  be  made  with  every 
engineering  investigation,  to  get  best  use  of  it. 

(a)  The  scientific  record  of  the  investigation.  This  must  be 
so  complete  as  to  enable  another  investigator  to  completely  check 
up,  repeat  and  further  extend  the  investigation.  It  thus  must 
contain  the  original  observations,  the  method  of  work,  apparatus 
and  facihties,  calibrations,  information  on  the  limits  of  accuracy 
and  reliabihty,  sources  of  error,  methods  of  calculation,  etc.,  etc. 

It  thus  is  a  lengthy  report,  and  as  such  will  be  read  by  very  few, 
if  any,  except  other  competent  investigators,  but  is  necessary 
as  the  record  of  the  work,  since  without  such  report,  the  work 
would  be  lost,  as  the  conclusions  and  results  could  not  be  checked 
up  if  required. 

This  report  appeals  only  to  men  of  the  same  character  as  the 
one  who  made  the  investigation,  and  is  essentially  for  record 
and  file. 

(b)  The  general  engineering  report.  It  should  be  very  much 
shorter  than  the  scientific  report,  should  be  essentially  of  the 
nature  of  a  syllabus  thereof,  avoid  as  much  as  possible  complex 
mathematical  and  theoretical  considerations,  but  give  all  the 
engineering  results  of  the  investigation,  in  as  plain  language  as 
possible.  It  would  be  addressed  to  administrative  engineers, 
that  is,  men  who  as  engineers  are  capable  of  understanding  the 
engineering  results  and  discussion,  but  have  neither  time  nor 
familiarity  to  follow  in  detail  through  the  investigation,  and  are 
not  interested  in  such  things  as  the  original  readings,  the  discus- 
sion of  methods,  accuracy,  etc.,  but  are  interested  only  in  the 
results. 

This  is  the  report  which  would  be  read  by  most  of  the  men 
interested  in  the  matter.  It  would  in  general  be  the  form  in 
which  the  investigation  is  communicated  to  engineering  societies 
as  paper,  with  the  scientific  report  relegated  into  an  appendix  of 
the  paper. 


292  ENGINEERING  MATHEMATICS. 

(c)  The  general  report.  This  should  give  the  results,  that  is, 
explain  what  the  matter  is  about,  in  plain  and  practically  non- 
technical language,  addressed  to  laymen,  that  is,  non-engineers. 
In  other  words,  it  should  be  understood  by  any  intelligent  non- 
technical man. 

Such  general  report  would  be  materially  shorter  than  the 
general  engineering  report,  as  it  would  omit  all  details,  and 
merely  deal  with  the  general  problem,  purpose  and  solution. 

In  general,  it  is  advisable  to  combine  all  three  reports,  by 
having  the  scientific  record  preceded  by  the  general  engineering 
report,  and  the  latter  preceded  by  the  general  report.  Roughly, 
the  general  report  would  usually  have  a  length  of  20  to  40  per 
cent  of  the  general  engineering  report,  the  latter  a  length  of  10 
to  25  per  cent  of  the  complete  scientific  record. 

The  bearing  of  the  three  classes  of  reports  may  be  understood 
by  illustration  on  an  investigation  which  appears  of  commercial 
utility,  and  therefore  is  submitted  for  industrial  development  to 
a  manufacturing  corporation;  the  financial  and  general  adminis- 
trative powers  of  the  corporations,  to  whom  the  investigation  is 
submitted,  would  read  the  general  report  and  if  the  matter 
appears  to  them  of  sufficient  interest  for  further  consideration, 
refer  it  to  the  engineering  department.  The  general  report  thus 
must  be  written  for,  and  intelligible  to  the  non-engineering 
administrative  heads  of  the  organization.  The  administrative 
engineers  of  the  engineering  department  then  peruse  the  general 
engineering  report,  and  this  report  thust  must  be  an  engineering 
report,  but  general  and  not  require  the  knowledge  of  the  specialist 
in  the  particular  field.  If  then  the  conclusion  derived  by  the 
administrative  engineers  from  the  reading  of  the  general  engineer- 
ing report  is  to  the  effect  that  the  matter  is  worth  further  con- 
sideration, then  they  refer  it  to  the  specialists  in  the  field  covered 
by  the  investigation,  and  to  the  latter  finally  the  scientific  record 
of  the  investigation  appeals  and  is  studied  in  making  final  report 
on  the  work. 

Inversely,  where  nothing  but  a  lengthy  scientific  report  is 
submitted,  as  a  rule  it  will  be  referred  to  the  engineering  depart- 
ment, and  the  general  engineer,  even  if  he  could  wade  through 
the  lengthy  report,  rarely  has  immediately  time  to  do  so,  thus 
lays  it  aside  to  study  sometime  at  his  leisure — and  very  often 
this  time  never  comes,  and  the  entire  matter  drops,  for  lack  of 
proper  representation. 


NUMERICAL  CALCULATIONS.  293 

Thus  it  is  of  the  utmost  importance  for  the  engineer  and  the 

.  scientist,  to  be  able  to  present  the  results  of  his  work  not  only  by 

elaborate  and  lengthy  scientific  report,  but  also  by  report  of 

moderate  length,  intelligible  without  difficulty  to  the  general 

engineer,  and  by  short  statement  intelhgible  to  the  non-engineer. 

d.  Reliability  of  Numerical  Calculations. 

172.  The  most  important  and  essential  requirement  of 
numerical  engineering  calculations  is  their  absolute  reliability. 
When  making  a  calculation,  the  most  brilliant  ability,  theo- 
retical knowledge  and  practical  experience  of  an  engineer  are 
made  useless,  and  even  worse  than  useless,  by  a  single  error  in 
an  important  calculation. 

Reliability  of  the  numerical  calculation  is  of  vastly  greater 
importance  in  engineering  than  in  any  other  field.  In  pure 
mathematics  an  error  in  the  numerical  calculation  of  an 
example  which  illustrates  a  general  proposition,  does  not  detract 
from  the  interest  and  value  of  the  latter,  which  is  the  main 
purpose;  in  physics,  the  general  law  which  is  the  subject  of 
the  investigation  remains  true,  and  the  investigation  of  interest 
and  use,  even  if  in  the  numerical  illustration  of  the  law  an 
error  is  made.  With  the  most  brilHant  engineering  design, 
however,  if  in  the  numerical  calculation  of  a  single  structural 
member  an  error  has  been  made,  and  its  strength  thereby  calcu- 
lated wrong,  the  rotor  of  the  machine  flies  to  pieces  by  centrifugal 
forces,  or  the  bridge  collapses,  and  with  it  the  reputation  of  the 
engineer.  The  essential  difference  between  engineering  and 
purely  scientific  caclulations  is  the  rapid  check  on  the  correct- 
ness of  the  calculation,  which  is  usually  afforded  by  the  per- 
formance of  the  calculated  structure — but  too  late  to  correct 
errors. 

Thus  rapidity  of  calculation,  while  by  itself  useful,  is  of  no 
value  whatever  compared  with  reUability — that  is,  correctness. 

One  of  the  first  and  most  important  requirements  to  secure 
reliability  is  neatness  and  care  in  the  execution  of  the  calcula- 
tion. If  the  calculation  is  made  on  any  kind  of  a  sheet  of 
paper,  with  lead  pencil,  with  frequent  striking  out  and  correct- 
ing of  figures,  etc.,  it  is  practically  hopeless  to  expect  correct 
results  from  any  more  extensive  calculations.     Thus  the  work 


293a  ENGINEERING  MATHEMATICS. 

should  be  done  with  pen  and  ink,  on  white  ruled  paper;  if 
changes  have  to  be  made,  they  should  preferably  be  made  by 
erasing,  and  not  by  striking  out.  In  general,  the  appearance  of 
the  work  is  one  of  the  best  indications  of  its  reliabihty.  The 
arrangement  in  tabular  form,  where  a  series  of  values  are  calcu- 
lated, offers  considerable  assistance  in  improving  the  reUability. 

173.  Essential  in  all  extensive  calculations  is  a  complete 
system  of  checking  the  results,  to  insure  correctness. 

One  way  is  to  have  the  same  calculation  made  independently 
by  two  different  calculators,  and  then  compare  the  results. 
Another  way  is  to  have  a  few  points  of  the  calculation  checked 
by  somebody  else.  Neither  way  is  satisfactory,  as  it  is  not 
always  possible  for  an  engineer  to  have  the  assistance  of  another 
engineer  to  check  his  work,  and  besides  this,  an  engineer  should 
and  must  be  able  to  make  numerical  calculations  so  that  he  can 
absolutely  rely  on  their  correctness  without  somebody  else 
assisting  him. 

In  any  more  important  calculations  every  operation  thus 
should  be  performed  twice,  preferably  in  a  different  manner. 
Thus,  when  multiplying  or  dividing  by  the  slide  rule,  the  multi- 
plication or  division  should  be  repeated  mentally,  approxi- 
mately, as  check;  when  adding  a  column  of  figures,  it  should  be 
added  first  downward,  then  as  check  upward,  etc. 

Where  an  exact  calculation  is  required,  first  the  magnitude 
of  the  quantity  should  be  estimated,  if  not  already  known, 
then  an  approximate  calculation  made,  which  can  frequently 
be  done  mentally,  and  then  the  exact  calculation;  or,  inversely, 
after  the  exact  calculation,  the  result  may  be  checked  by  an 
approximate  mental  calculation. 

Where  a  series  of  values  is  to  be  calculated,  it  is  advisable 
first  to  calculate  a  few  individual  points,  and  then,  entirely 
independently,  calculate  in  tabular  form  the  series  of  values, 
and  then  use  the  previously  calculated  values  as  check.  Or, 
inversely,  after  calculating  the  series  of  values  a  few  points 
should  independently  be  calculated  as  check. 

When  a  series  of  values  is  calculated,  it  is  usually  easier  to 
secure  reliability  than  when  calculating  a  single  value,  since 
in  the  former  case  the  different  values  check  each  other.  There- 
fore it  is  always  advisable  to  calculate  a  number  of  values, 
that  is,  a  short  curve  branch,  even  if  only  a  single  point  is 


NUMERICAL  CALCULATIONS.  2936 

required.  After  calculating  a  series  of  values,  they  are  plotted 
as  a  curve  to  see  whether  they  give  a  smooth  curve.  If  the 
entire  curve  is  irregular,  the  calculation  should  be  thrown  away, 
and  the  entire  work  done  anew,  and  if  this  happens  repeatedly 
with  the  same  calculator,  the  calculator  is  advised  to  find 
another  position  more  in  agreement  with  his  mental  capacity. 
If  a  single  point  of  the  curve  appears  irregular,  this  points  to 
an  error  in  its  calculation,  and  the  calculation  of  the  point  is 
checked;  if  the  error  is  not  found,  this  point  is  calculated 
entirely  separately,  since  it  is  much  more  difficult  to  find  an 
error  which  has  been  made  than  it  is  to  avoid  making  an 
error. 

174.  Some  of  the  most  frequent  numerical  errors  are: 

1.  The  decimal  error,  that  is,  a  misplaced  decimal  point. 
This  should  not  be  possible  in  the  final  result,  since  the  magni- 
tude of  the  latter  should  by  judgment  or  approximate  calcula- 
tion be  known  sufficiently  to  exclude  a  mistake  by  a  factor  10. 
However,  under  a  square  root  or  higher  root,  in  the  exponent 
of  a  decreasing  exponential  function,  etc.,  a  decimal  error  may 
occur  without  affecting  the  result  so  much  as  to  be  immediately 
noticed.  The  same  is  the  case  if  the  decimal  error  occurs  in  a 
term  which  is  relatively  small  compared  with  the  other  terms, 
and  thereby  does  not  affect  the  result  very  much.  For  instance, 
in  the  calculation  of  the  induction  motor  characteristics,  the 
quantity  ri^-\-sHi^  appears,  and  for  small  values  of  the  slip  s, 
the  second  term  s'^Xi^  is  small  compared  with  ri^,  so  that  a 
decimal  error  in  it  would  affect  the  total  value  sufficiently  to 
make  it  seriously  wrong,  but  not  sufficiently  to  be  obvious. 

2.  Omission  of  the  factor  or  divisor  2. 

3.  Error  in  the  sign,  that  is,  using  the  plus  sign  instead  of 
the  minus  sign,  and  inversely.  Here  again,  the  danger  is 
especially  great,  if  the  quantity  on  which  the  wrong  sign  is 
jsed  combines  with  a  larger  quantity,  and  so  does  not  affect 
the  result  sufficiently  to  become  obvious. 

4.  Omitting  entire  terms  of  smaller  magnitude,  etc. 


APPENDIX  A. 

NOTES  ON  THE  THEORY  OF  FUNCTIONS. 

A.  General  Functions. 

175.  The  most  general  algebraic  expression  of  powers  of 
X  and  y, 

■^(a^;2/)  =  («oo +  001-^  +  002x2  +  .  .  .)  +  (aio +aiix  +  ai2.r2  + .  .  '.)y 

+  (020+021-1 +  022.^2  +  .  •  -^2/2  +  -  •  • 

+  (ar,o+a„iJ:  +  o„2-c2  +  - •  -  )?/"=0,        ....     (I) 

is  the  implicit  analytic  function.  It  relates  y  and  x  so  that  to 
every  value  of  x  there  correspond  n  values  of  y,  and  to  every 
value  of  y  there  correspond  m  values  of  x,  if  m  is  the  exponent 
of  the  highest  power  of  x  in  (1). 

Assuming  expression  (1)  solved  for  y  (which  usually  cannot 
be  carried  out  in  final  form,  as  it  requires  the  solution  of  an 
equation  of  the  nth  order  in  y,  with  coefficients  which  are 
expressions  of  x),  the  explicit  analytic  function, 

y=f{x), (2) 

is  obtained.  Inversely,  solving  the  imphcit  function  (1)  for 
X,  that  is,  from  the  explicit  function  (2),  expressing  x  as 
function  of  y,  gives  the  reverse  function  of  (2);  that  is 

x=/i(2/) (3) 

In  the  general  algebraic  function,  in  its  implicit  form  (1), 
or  the  explicit  form  (2),  or  the  reverse  function  (3),  x  and  y 
are  assumed  as  general  numbers;  that  is,  as  complex  quan- 
tities; thus, 

x  =  xi+jx2;  1 

(4) 

y=yi+m>  J 

and  likewise  are  the  coefficients  ooo,  oqi  .  .  .  a„„,. 

2»4 


APPENDIX  A.  295 

If  all  the  coefficients  a  are  real,  and  x  is  real,  the  corre- 
sponding n  values  of  y  are  either  real,  or  pairs  of  conjugate 
complex  imaginary  quantities:  y\+jy2  and  yi  —  jy2- 

176.  For  n  =  l,  the  implicit  function  (1),  solved  for  y,  gives 
the  rational  function, 

aoo+aoi.r+ao23'^  +  .  .  . 
^~aio  +  aiia:  +  a]2.r-  +  .  .    ' ^^ 

and  if  in  this  function  (5)  the  denominator  contains  no  x,  the 
integer  function, 

y  =  ao+aix+a2X^-\  .  .  .+a„,x"',     ,     .     .     .     (6) 

is  obtained. 

For  n  =  2,  the  implicit  function  (1)  can  be  solved  for  ?/  as  a 
quadratic  equation,  and  thereby  gives 

—  (0)0+ 01,3^ +  ai;x'  +  ---)d: 

y  2(ajo  +  a2iX  +  a22x2  +  ...)  '  ^'^ 

that  is,  the  explicit  form  (2)  of  equation  (1)  contains  in  this 
case  a  square  root. 

For  n>2,  the  explicit  form  y=f(x)  either  becomes  very 
complicated,  for  n  =  3  and  n  =  4,  or  cannot  be  produced  in 
finite  form,  as  it  requires  the  solution  of  an  equation  of  more 
than  the  fourth  order.  Nevertheless,  y  is  still  a  function  of 
X,  and  can  as  such  be  calculated  by  aj^proximation,  etc. 

To  find  the  value  yi,  which  by  function  (1)  corresponds  to 
x  =  Xi,  Taylor's  theorem  offers  a  rapid  approximation.  Sub- 
stituting xi  in  function  (1)  gives  an  expression  which  is  of 
the  nth  order  in  y,  thus:  F(x\y),  and  the  problem  now  is  to 
find  a  value  y\,  which  makes  F{_x\,yi)=Q. 

However, 

N     r..        N     ,  dF(xi,y)     h^d^F(xuy) 
F{x,,y,)^F(x,,y)+h^-^^+~      ^^^y  +.  .  .  ,   .     (8) 

where  h^yi  —  y  is  the  difference  between  the  correct  value  yi 
and  any  chosen  value  y. 


296 


ENGINEERING  MA  THEM  A  TICS. 


Neglecting  the  higher  orders  of  the  small  quantity  h,  in 
C8),  and  considering  that  F(xi,7/i)  =0,  gives 


h=- 


F(xi,y) 

dF{x,,yy 

dy 


(y) 


and  herefrom  is  obtained  yi  =  y+h,  as  first  approximation. 
Using  this  value  of  yi  as  y  in  (9)  gives  a  second  approximation, 
which  usually  is  sufficiently  close. 

177.  New  functions  arc  defined  by  the  integrals  of  the 
analytic  functions  (1)  or  (2),  and  by  their  reverse  functions. 
They  are  called  Abelian  integrals  and  Ahelian  functions. 

Thus  in  the  most  general  case  (1),  the  expUcit  function 
corresponding  to  (1)  being 


y=f{x),  .    . 

2=  I  fi^)dx, 


(2) 


the  integral, 

then  is  the  general  Abelian  integral,  and  its  reverse  function, 


a:  =  0(2), 

the  general  Abelian  function. 

(a)  In  the  case,  n  =  l,  function  (2)  gives  the  rational  function 
(5),  and  its  special  case,  the  integer  function  (6). 

Function  (G)  can  be  integrated  by  powers  of  x.  (5)  can  be 
resolved  into  partial  fractions,  and  thereby  leads  to  integrals 
of  the  following  forms : 

(1)  Jwx; 

(2)  P^; 


^^^     j   [x-a)m' 


(10) 


APPENDIX  A.  297 

Integrals  (10),  (1),  and  (3)  integrated  give  rational  functions, 
(10),  (2)  gives  the  logarithmic  function  log  {x—a),  and  (10),  (4) 
the  arc  function  arc  tan  x. 

As  the  arc  functions  are  logarithmic  functions  with  complex 
imaginary  argimient,  this  case  of  the  integral  of  the  rational 
function  thus  leads  to  the  logarithmic  function,  or  the  loga- 
rithmic integral,  which  in  its  simplest  form  is 

2=  (  y  =  logx, (11) 

and  gives  as  its  reverse  function  the  exponential  function, 

x=e' (12) 

It  is  expressed  by  the  infinite  series, 

z^     z^     z^ 
-  =  l+.e+j^  +  |3+|^  + (13) 

as  seen  in  Chapter  II,  paragi'aph  53. 

178.  6.  In  the  case,  n=2,  function  (2)  appears  as  the  expres- 
sion (7),  which  contams  a  square  root  of  some  power  of  x.  Its 
first  part  is  a  rational  function,  and  as  such  has  already  been 
discussed  in  a.    There  remains  thus  the  integral  function, 


■/ 


Vbo+biX  +  b2X-  +  .  .  .+bpXP 

Co+ClX+C2X~  +  . 


2=      —""^rTT-TTT^. ~d.i:    •     ■     •     (14) 


This  expression  (14)  leads  to  a  series  of  important  functions. 
(1)  Forp  =  l  or  2, 

r  Vbo+biX  +  b2X^    ,  ^   . 

z=  I  — \ 5- —  dx (15) 

J    Co+CiX+C2X^  +  .  .  •  ^       ' 

By  substitution,  resolution  into  partial  fractions,  and 
separation  of  rational  functions,  this  integral  (11)  can  be 
reduced  to  the  standard  form, 

dx 
In  the  case  of  the  minus  sign,  this  gives 


'-^T^^^ (^6) 


;dx 


298 


ENGINEERING  MATHEMATICS. 


and  as  reverse  functions  thereof,  there  are  obtained   the  trigo- 
nometric  functions. 

x  =  sin  z,  ] 

, ,         (IS) 

vl  — j:-  =  cos  z.  J 
In  the  case  of  the  plus  sign,  integral  (16)  gives 

—  =  —  log  I  vT+x^ — X I  =  arc  sinh  x^     .    (19) 

V  1  +x2 

and  reverse  functions  thereof  are  the  hyperbolic  functions, 

c  +  2_   £-Z  _  1 

:z =sinh0; 


x  =  - 


P  +  Z_|_   -  — z 

VT+X^  = ?;^ =  cosh  . 


(20) 


The  trigonometric  functions  are  expressed  by  the  series : 


(21) 


2;3       2^5       2^7 

sin  z  =  z—T-r +7^  —  7^ +  .  .  .  ; 

2.2       04        ^6 

COS  0=1  —  rT+TT-"T7^  +  -  •  •  . 
2         4        0 


as  seen  in  Chapter  II,  paragraph  58, 

The  hyperbolic  functions,  by  substituting  for  £+^  and  £~' 
the  series  (13),  can  be  expressed  by  the  series: 


y3  ;5 


sinh  3  =  2+7^+7^+7;=-  +  .  .  .  ; 
1^    1^    \l 

Z~        Z'^       2® 
C0sh2=l+j7r+j7-+j7r+.  .  .  . 

I       r 


(22) 


179.  In  the  next  case,  p  =  3  or  4, 


z  = 


X+b2X^+b3X^+b4X'*' 


+  CiX+C2X^  +  .  .  ■ 


dx,     .     .     (23) 


already  leads  beyond  the  elementary  functions,  that  is,  (23) 
cannot  be  integrated  by  rational,  logarithmic  or  arc  functions, 


APPENDIX  A. 


299 


but  gives  a  new  class  of  functions,  the  elliptic  integrals,  and 
their  reverse  functions,  the  elliptic  functions,  so  called,  because 
they  bear  to  the  ellipse  a  relation  similar  to  that,  which  the 
trigonometric  functions  bear  to  the  circle  and  the  hyperbolic 
functions  to  the  equilateral  hyperbola. 

The  integral  (23)  can  be  resolved  into  elementary  functions, 
and  the  three  classes  of  elliptic  integrals : 


-i 


dx 


Vx{l-x){l-c^xy 
xdx 

vx[i-x)ii-cHy 

dx 

(x-b)  Vx{l-x)il-c^x) ' 


(24) 


(These  three  classes  of  integrals  may  be  expressed  in  several 
different  forms.) 

The  reverse  functions  of  the  elliptic  integrals  are  given  by 
the  elliptic  functions : 

v'j:  =  sin  am{u,c) 


y/l  —  x  =  cos  am(u,  c) ; 
Vl  —c^x  =  Jam(ii,  c); 


(25) 


known,  respectively,  as  sine-amplitude,  cosine-amplitude,  delta- 
amplitude. 

Elliptic  functions  are  in  some  respects  similar  to  trigo- 
nometric functions,  as  is  seen,  but  they  are  more  general, 
depending,  as  they  do,  not  only  on  the  variable  x,  but  also  on 
the  constant  c.  They  have  the  interesting  property  of  being 
doubly  periodic.  The  trigonometric  functions  are  periodic,  with 
the  periodicity  2;:,  that  is,  repeat  the  same  values  after  every 
change  of  the  angle  by  2?:.  The  elliptic  functions  have  two 
periods  pi  and  p2,  that  is. 


sin  am(u  +npi  +mp2,  c)  =sin  am{u,  c),  etc.; 


(26) 


hence,   increasing  the   variable  u  by  any  multiple  of  either 
period  pi  and  p2,  repeats  the  same  values. 


300  ENGINEERING  MATHEMATICS. 

The  two  periods  are  given  by  the  equations, 


r'  dx 

J,  2\  j:(l-~x){i-c^x) 


(27) 


!vx(l-x)(l-c^x) 

i8o.  ElUptic  functions  can  be  expressed  as  ratios  of  two 
infinite  series,  and  these  series,  whicli  fonn  the  numerator  and 
the  denominator  of  the  elliptic  function,  are  called  theta  func- 
tions and  expressed  by  the  symbol  6,  thus 


sin  am{u,  c)  = 


cos 


-u 


e. 


-u 


Jam{u,  c)      =  a/ 1  -  c^—p^r- 


(28) 


and  the  four  6  functions  may  be  expressed  by  the  series : 
do{x)  =  1  -2q  cos  2j;  +2^4  eos  4x  -2^9  cos  6x  4-  -. 

di{x)  =291/4  gin  X  -2^9/4  gin  Sx+2qT  sin  5x-  +  . 

25 

/92(x)  =2^1/4  COS  x  +2g9/4  COS  3x  +2^  4  cos  5x  +  .  . 
63{x)  =  1  +22  cos  2x  +2^  cos  4x  +2^9  cos  6x  + .  .  . 


1 


(29) 


where 


1  .    ?>'> 

Q=£'^    and     a  =  ir.^—. 

'    Pi 


(30) 


In  the  case  of  integral  function  (14),  where  p>4,  similar 
integrals   and  their  reverse  functions   appear,   more   complex 


APPENDIX  A.  301 

than  the  elUptic  functions,  and  of  a  greater  number  of  periodici- 
ties. They  are  called  hyperelliptic  integrals  and  hyperelliptic 
functions,  and  the  latter  are  again  expressed  by  means  of  auxil- 
iary' functions,  the  hyperelliptic  0  functions. 

i8i.  Many  problems  of  physics  and  of  engineering  lead  to 
elliptic  functions,  and  these  functions  thus  are  of  considerable 
importance.  For  instance,  the  motion  of  the  pendulum  is 
expressed  by  elliptic  functions  of  time,  and  its  period  thereby 
is  a  function  of  the  ampHtude,  increasing  with  increasing  ampli- 
tude; that  is,  in  the  so-called  "second  pendulum,"  the  time  of 
one  swing  is  not  constant  and  equal  to  one  second,  but  only 
approximately  so.  This  approximation  is  very  close,  as  long 
as  the  amplitude  of  the  swing  is  very  small  and  constant,  but 
if  the  amplitude  of  the  swing  of  the  pendulum  varies  and 
reaches  large  values,  the  time  of  the  swing,  or  the  period  oi 
the  pendulum,  can  no  longer  be  assumed  as  constant  and  an 
exact  calculation  of  the  motion  of  the  pendulum  by  elliptic 
functions  becomes  necessary. 

In  electrical  engineering,  one  has  frequently  to  deal  with 
oscillations  similar  to  those  of  the  pendulum,  for  instance, 
in  the  hunting  or  surging  of  synchronous  machines.  In 
general,  the  frequency  of  oscillation  is  assumed  as  constant, 
but  where,  as  in  cumulative  hunting  of  synchronous  machines, 
the  amplitude  of  the  swing  reaches  large  values,  an  appreciable 
change  of  the  period  must  be  expected,  and  where  the  hunting 
is  a  resonance  effect  with  some  other  periodic  motion,  as  the 
engine  rotation,  the  change  of  frequency  with  increase  of 
amplitude  of  the  oscillation  breaks  the  complete  resonance  and 
thereby  tends  to  limit  the  amplitude  of  the  swing. 

182.  As  example  of  the  application  of  elliptic  integrals,  may 
be  considered  the  determination  of  the  length  of  the  arc  of  an 
ellipse. 

Let  the  ellipse  of  equation 

a2  +  62-l. ^^^) 

be  represented  in  Fig.  93,  with  the  circumscribed  circle, 

a:2  +  ?/2  =  a2 (32) 


302 


ENGINEERING  MATHEMATICS. 


To  every  point  P  =  x,  y  of  the  ellipse  then  corresponds  a 
pohit  P\=x,  yi  on  the  circle,  which  has  the  same  abscissa  x, 
and  an  angle  d  =  AOP\. 

The  arc  of  the  ellipse,  from  A  to  P,  then  is  given  by  the 
integral, 

where 


0  =  sm2^=l-)      and     c=- 

\a  a 


.     (34) 


is  the  eccentricity  of  the  ellipse. 


Fig.  93.     Rectification  of  Ellipse. 

Thus  the  problem  leads  to  an  elliptic  integral  of  the  first 
and  of  the  second  class. 

For  more  complete  discussion  of  the  elliptic  integrals  and 
the  elliptic  functions,  reference  must  be  made  to  the  text-books 
of  mathematics. 

B.  Special  Functions. 

183.  Numerous  special  functions  have  been  derived  by  the 
exigencies  of  mathematical  problems,  mainly  of  astronomy,  but 
in  the  latter  decades  also  of  physics  and  of  engineering.  Some 
of  them  have  already  been  discussed  as  special  cases  of  the 
general  Abelian  integral  and  its  reverse  function,  as  the  expo- 
nential, trigonometric,  hyperbolic,  etc.,  functions. 


APPENDIX  A.  303 

Functions  may  be  represented  by  an  infinite  series  of  terms; 
that  is,  as  a  sum  of  an  infinite  number  of  terms,  which  pro- 
gressively decrease,  that  is,  approach  zero.  The  denotation  of 
the  terms  is  commonly  represented  by  the  summation  sign  H. 

Thus  the  exponential  functions  may  be  written,  when 
defining, 

|0  =  1;         |m  =  1x2x3x4X.  .  .Xn, 

as 

e'  =  l+x+g  +  i3+...=.S.j^ (35) 

which  means,  that  terms  ,—  are  to  be  added  for  all  values  of  n 

'  \n 

from  n  =  0  to  n  =  oc  . 

The  trigonometric  and  hyperbolic  functions  may  be  written 
in  the  form : 

r3        j-5        >-7  -JO  ^2n  +  l 

3i„,.,,_+-j^__+....S»(-lV|j^;.     (36) 


Y»2       jA       j-6  y^  ^2n 

cos.=l-g+g-^  +  .    .  =  S..r-l)..i^:    •     •     (37) 

X'^     x^     x^  ^     x^"  "^^ 

sinhx^x+T77+TF+T^  +  - • -^^^ ..     , -.  :    •     •     •     (38) 


|3     |5  '  |7  0    |2n  +  l 


x^    x^    x^  ^  x-'^ 


C0shx=l+nr+rr+T7r  +  .  .  .  =  2'»r— (39) 

|2     |4     jb  0    Izi^ 

Functions  also  may  be  expressed  by  a  series  of  factors; 
that  is,  as  a  product  of  an  infinite  series  of  factors,  which  pro- 
gressively approach  unity.  The  product  series  is  commonly 
represented  by  the  symbol  ["J. 

Thus,  for  instance,  the  sine  function  can  be  expressed  in  the 
form, 

™-  =  -(l-^^)(l-^.)(l-£.)--4(l-;|^).     (-10) 

184.  Integration  of  known  functions  frequently  leads  to  new 
functions.     Thus   from   the  general   algebraic   functions   were 


304  ENGINEERING  MATHEMATICS. 

derived  the  Abelian  functions.  In  physics  and  in  engineering, 
integration  of  special  functions  in  this  manner  frequently  leads 
to  new  special  functions. 

For  instance,  in  the  study  of  the  propagation  through  space, 
of  the  magnetic  field  of  a  conductor,  in  wireless  telegraphy, 
lightning  protection,  etc.,  we  get  new  functions.  If  i=f  (t) 
is  the  current  in  the  conductor,  as  function  of  the  time  t,  at  a 
distance  x  from  the  conductor  the  magnetic  field  lags  by  the 

X 

time  ^1  =  -^,  where  S  is  the  speed  of  propagation  (velocity  of 

light).  Since  the  field  intensity  decreases  inversely  propor- 
tional to  the  distance  x,  it  thus  is  proportional  to 


f{'-i) 


(41) 


and  the  total  magnetic  flux  then  is 


=  j  ydx 


dx (42) 


If  the  current  is  an  alternating  current,  that  is,  f  (t)  a 
trigonometric  function  of  time,  equation  (42)  leads  to  the 
functions, 


f  sin  X 

u=   I ax: 

J     ^ 

/cos  X   , 
dx. 
X 


(43) 


If  the  current  is  a  direct  current,  rising  as  exponential 
function  of  the  time,  equation  (42)  leads  to  the  function, 


/£^dx 


^=  I  -^- (44) 


APPENDIX  A. 


305 


Substituting  in  (43)  and  (44),  for  sin  x,  cos  x,  e'  thoir 
infinite  scries  (21)  and  (13),  and  then  integrating,  gives  the 
following: 


sin  X  x-^      J^      .r^ 

X  3|3     5|o     /|/ 


cosx 


.     (45) 


c  ^        2|2     4|4     6|6 

— r/.r  =  logx+x+:^  +-^  +.  .  . 
X  ^  2|2     3|3 

For  further  discussion  and  tables  of  these  functions  see 
"  Theory  and  Calculation  of  Transient  Electric  Phenomena  and 
Oscillations,"  Section  III,  Chapter  VIII,  and  Appendix. 

.8S.  If  y^fi.)  is  a  function  of  x,  and  .=//Wd.  =  ^(.) 

its  integral,  the  definite  integral,  Z=  I  f{x)dx,   is    no  longer 
a  function  of  x  but  a  constant, 

Z  =  c;6(6)-<^(a). 
For  instance,  if  y^c{x—nY,  then 

c{x—nY 


z=  I  cix— 


nydx=- 


and  the  definite  integral  is 


Z  = 


f}''- 


nYdx  =  ^{{h-n)^-ia-ny}. 


This  definite  integral  does  not  contain  x,  but  it  contains 
all  the  constants  of  the  function  f{x),  thus  is  a  function  of 
these  constants  c  and  n,  as  it  varies  with  a  variation  of  these 
constants. 

In  this  manner  new  functions  may  be  derived  by  definite 
integrals. 

Thus,  if 

y=f{x,u,v...) (46) 

is  a  function  of  x,  containing  the  constants  u,  v  . .  . 


306  ENGINEERING  MATHEMATICS. 

The  definite  integral, 

Z=£f{x,u,v...)dx, (47) 

is  not  a  function  of  .r,  but  still  is  a  function  of  m,  v  .  .  .  ,  and 
may  be  a  new  function. 

186.  For  instance,  let 

2/=£-Xj;«-l. (4g) 

then  the  integral, 

f(u)=  H-^x'^-'^dx, (49) 

is  a  new  function  of  u,  called  the  gamma  function. 

Some  properties  of  this  function  may  be  derived  by  partial 
integration,  thus : 

rCu  +  l)=ur{u); (50) 

if  n  is  an  integer  number, 

r{u)  =  (:u-l){u-2)..  .(u-n)r(u-n),     .     .     (51) 
and  since 

r{i)  =  i, (52) 

if  u  is  an  integer  number,  then, 

r{u)==\u-l.         (53) 

C.   Exponential,  Trigonometric  and  Hyperbolic  Functions. 

(a)  Functions  of  Real  Variables. 

187.  The  exponential,  trigonometric,  and  hyperbolic  func- 
tions are  defined  as  the  reverse  functions  of  the  integrals, 

rdx    , 
a.  u=   I— =  logx, (54) 

and:  x=s" (55) 

r   dx 

0.  u=  \  — =  arcsm  x;     .....     (56) 

J  V  1  —  x^ 


APPENDIX  A.  307 

and:  x  =  sinM, (57) 

Vl  —  x^  =  cos  u, (58) 

C.  ^.=  ^-^=-log|VTT^-x{;.     .     .     .     (59) 

£"  _  j-« 
and  x  = ^ — =sinhw;      ....     (60) 

vl+x2  = — =coshw (61) 

From  (57)  and  (58)  it  follows  that 

sin^  M+cos^  ?^  =  1 (62) 

From  (60)  and  (61)  it  follows  that 

cos^ /iw— sin  2/iM  =  l.         (63) 

Substituting  (  — j)  for  x  in  (56),  gives  (  — w)  instead  of  u, 
and  therefrom, 

sin  ( —  w)  =  —  sin  M (64) 

Substituting  (— w)  for  u  in   (60),  reverses  the  sign  of   x, 
that  is, 

sinh  (— w)  =  —  sinh  u.  .     ,     .     (65) 

Substituting  (—x)  for  x  in  (58)  and  (61),  does  not  change 
the  value  of  the  square  root,  that  is, 

cos  ( —  u)  =  cos  u, (66) 

cosh  (  —  m)=  cosh  w,       (67) 

Which  signifies  that  cos  u  and  cosh  u  are  even  functions,  while 
sin  u  and  sinh  u  are  odd  functions. 

Adding  and  subtracting  (60)  and  (61),  gives 

£±'*  =  cosh  M±  sinh  1/ (68) 


308  ENGINEERING  MATHEMATICS. 

(b)  Functions  of  Imaginary  Variables. 
i88.  Substituting,  in  (56)  and  (59),  x=  —jy,  thus  y  =  jx,  gives 

/dx  C     dx 


+  X2 


x  =  smu;  x  =  sinhw=- 


c-M ff— « 


Q  ' 


£«  +  £-« 


\/l+x^  =  cosu;  Vl+x2  =  cosh  u  =  — ^ ; 

hence,  ju  =  j  :^;|= .  hence,  ju  =  J  ^7f=r ^ 

i/  =  sinhyM  = -^ ;  y=smju',     .    .    .     (69) 

vTT^  =  coshjw  = :^ ;  Vl  — y^  =  cos  ju;    .    .     .     (70) 

Resubstituting  x  in  both 

sinh  ff^     £j"_  £-3"  £"_£-M     sin  m     .„_ 

a:  =  sinw  = r^-  =  — w-- ;  x  =  smh?i  =  — :z —  =  — -. — ;  (71) 


£"•  + 


Vl  —  x^  =  cos  w  =  cosh  ju  vl+x2  =  coshw  = ^^ — 

= 2 '  =cosyw.     .     (72) 

Adding  and  subtracting, 

£±/"  =  cos  u±j  sin  w  =  cosh  j7^±siuh  ju 
and  £±'*  =  cosh  it±sinhM  =  cos  jii=Fj  sin  ju.     .     .     (73) 

(c)  Functions  of  Complex  Variables 

189.  It  is: 

cu±jv=  c"£±j»^£«(cos  y±jsin  v);      .     .     .     (74) 


APPENDIX  A. 


309 


sin  iu±jv)  =sin  u  cos  jv±cos  u  sin  jv 

=  Sin  u  cosh  u  ±  7  cos  u  sinh  ?;  =  — ^ —  sin  u  ±  ] — 7^ —  cos  u ; 


(75) 


cos  {u  ±  jv)  =  cos  u  COS  /i'  =F  sin  u  sin  p 

=  cos  w  cosh  ?'T  jsin  u  sinh  v  =  — ^ — cos  wT  / — ^ —  sin 


(76) 


sinh(M±p) 


pM±;t) ^  —  uTjv       g-« g— M 


.£"+£ 


"  -L  f  — " 


-;y — cosvij^ — sinu 
■■  sinh  u  cos  v  ±  /  cosh  u  sin  ?; ; 


(77) 


.  X  «.  —  li 


cosh(w  ±  ju)  = r~o =  — o —  ^"s  V  ±  / — ^ — sin  v 


■■  cosh  u  cos  v±j  sinh  i*  sin  u; 


etc. 


(7S) 


(d)  Relations. 

190.  From  the  preceding  equations  it  thus  follows  that  the 
three  functions,  exponential,  trigonometric,  and  hyperbolic, 
are  so  related  to  each  other,  that  any  one  of  them  can  be 
expressed  by  any  other  one,  so  that  when  allowing  imaginary 
and  complex  imaginary  variables,  one  function  is  sufficient. 
As  such,  naturally,  the  exponential  function  would  generally 
be  chosen. 

Furthermore,  it  follows,  that  all  functions  with  imaginary 
and  complex  imaginary  variables  can  be  reduced  to  functions 
of  real  variables  by  the  use  of  only  two  of  the  three  classes 
of  functions.  In  this  case,  the  exponential  and  the  trigono- 
metric functions  would  usually  be  chosen. 

Since  functions  with  complex  imaginary  variables  can  be 
resolved  into  functions  with  real  variables,  for  their  calculation 
tables  of  the  functions  of  real  variables  are  sufficient. 

The  relations,  by  which  any  function  can  be  expressed  by 
any  other,  are  calculated  from  the  preceding  paragraph,  by 
the  following  equations : 


310  ENGINEERING  MATHEMATICS. 

£*'*  =  cosh  u±sinh  m  =  cos  ;m^  j  sin  ju; 
£='='^  =  cos  i;  ± y  sin  v  =  cosh  jv ±  j  sinh  jv; 

sinh  ju     £'"—  £""* 


sin  u  = 


J 


2/       ' 


sin  jv  =  j  sinh  r  =  j 


sin  (m  ± ;'?;)  =  sin  u  cosh  v  ±  j  cos  m  sinh  v 

= ^ —  sin  u±j  — -^ cos u ; 


cos  u  =  cosh  ju  = 
cos  jv  =  cosh  I' 


£<U   -|_    £-7" 

2  ' 


COS  {u±jv)  =  cos  w  cosh  vT  f  sin  li  sinh  v 

gV £— W  e« £— V 

=  — ;^ —  cos  wT  ? — i^ —  sin  u; 


sinh  u  = 


£"—  £   "    sin  ju 


(a) 


sinh  J  y  =  /  sin  v  = 


£?"—    £-J« 


sinh  (w  ±  p)  =  sinh  u  cos  y  ±  /  cosh  u  sin  v 

gU £■""  £U_j_£— U 

=  — ^ —  cos  v±j ;^ sin  v; 


£«+£-« 

cosh  w  =  — ^ =  cos  ju; 


cosh  jv  =  cos  I'  = 


£jv  _|_  c-;v 


cosh  (u  ±  jv)  =  cosh  u  cos  v  ±  /  sinh  u  sin  v 


£«+£"" 


r« c— U 


COS  r  ±y- 


sin  v. 


APPENDIX  A.  311 

And  from  (h)  and  (d),  respectively  (c)  and  (e),  it  follows  that 
sinh  {u ± jv)  =  i sin  i±v— fti)  =  ±j sin  iv ±  ju) ; 


cosh  (m±;i')  =  cos  (i'=F;w). 

Tables  of  the  exponential  functions  and  their  logarithms, 
and  of  the  hyperbolic  functions  with  real  variables,  are  given 
in  the  following  Appendix  B. 


APPENDIX   B. 

TWO    TABLES    OF    EXPONENTIAL    AND    HYPERBOLIC 

FUNCTIONS. 

Table  I. 


£  =  2.7183, 


log  £  =  0.4343. 


I 

X10-' 

XlO-2 

xio-i 

XI 

1.0 

0.999 

0.990 

0.905 

0.368 

1.2 

0.988 

0.887 

0.301 

1.4 

0.986 

0.869 

0.247 

1.6 

0.984 

0.852 

0.202 

1.8 

0.982 

0.835 

0.165 

2.0 

0.998 

0.980 

0.819 

0.135 

2.5 

0.975 

0.779 

0.082 

3.0 

0.997 

0.970 

0.741 

0.050 

3.5 

0.966 

0.705 

0.030 

4.0 

0.996 

0.961 

0.670 

0.018 

4.5 

0.956 

0.638 

0.011 

5.0 

0.995 

0.951 

0.607 

0.007 

6 

0.994 

0.942 

0.549 

0.002 

7 

0.993 

0.932 

0.497 

0.001 

8 

0.992 

0.923 

0.449 

0.000 

9 

0.991 

0.914 

0.407 

10 

0 .  990 

0.905 

0.368 

312 


APPENDIX  B. 


313 


Table  IT. 


EXPONENTIAL  AND   HYPERBOLIC   FUNCTIONS. 

£  =  2.718282 ~ 2.7183,  log  e  =  0.4342945  ~ 0.4343. 

cosh  I  =  *je  +  ^ +  £-='!,  sinh  X  =  h\£  +  ' -  £-^\. 


...  1 

434 

435 

0 

0 

0 

0.1 

43 

43 

0.2 

87 

87 

0.3 

130 

130 

0.4 

174 

174 

0.5 

217 

217 

0.6 

261 

261 

0.7 

304 

304 

0.8 

347 

348 

0.9 

391 

391 

1.0 

434 

435 

0 


0.001 
0.002 
0.003 
0.004 


0.005 


0.006 
0.007 
0.008 
0.009 


0.010 


0.012 
0.014 
0.016 
0.018 


0.020 


0.025 
0.030 
0.035 
0.040 
0.045 


0.050 


0.06 
0.07 
0.08 
0.09 


0.10 


0.12 
0.14 
0.16 
0.18 


0.20 


log£  +  ^ 


0 


0 . 000434 
0.000869 
0.001303 
0.00173 


0.002171 


0.002606 
0.003040 
0 . 003474 
0 . 003909 


0.004343 


0.005212 
0.006080 
0 . 006949 
0.00781 


0.008686 


0.010857 
0.013029 
0.015200 
0.017372 
0.019543 


0.021715 


0.026058 
0.030401 
0.034744 
0.039086 


0.043429 


0.052115 
0.060801 
0.069487 
0.078173 


0.086859 


^log 


434 
435 
434 
434 

434 

435 

434 
434 
435 

434 


log  £ 


0 


9.999566 
9.999131 
9 . 99869 
9 . 998263 


9.997829 


9.997394 
9 . 996960 
9 . 996526 
9.996091 


9.99565: 


9.994788 
9.993920 
9.993051 
9.992183 


9.991314 


9.989143 

9. 

9 . 984800 

9 .  98262: 

9.980457 


9.978285  1.0512 


9.973942 
9.969599 
9.965256 
9.960914 


9.956571 


9.947885 
9.939199 
9.930513 
9.921827 


9.913141 


£  +  ' 


1 


1.00100 
1.00200 
1.00301 
1.00401 


1.00501 


1 . 00602 
1 . 00702 
1.00803 
1 . 00904 


1.01005 


1.01207 
1.01410 
1.01613 
1.01816 


1.02020 


28  1 


1.02531 
1 . 03046 
1.03562 
04081 
1 . 04603 


1.06184 
1.07251 
1.08329 
1.09417 


1.10516 


1.12750 
1.15027 
1.17351 
1.19721 


1.22140 


1 


0 . 99900 
0 . 99800 
0 . 99700 
0.99601 


0.99501 


0.99402 
0.99302 
0 . 99203 
0.99104 


0.99005 


0.98807 
0.9861U 
0.98413 
0.98216 


0.98020 


0.97531 
0.970451 
0.96561 
0 . 96079 
0.9560011 


0.95123 


0.9417 
0.93239 
0.92312 
0.91393 


0.90484 


0.88692 
0 . 86936 
0.85214 
0.83527 


0.81873 


cosh  X     sinh  x 


1 


1.00000 
1.00000 
1.00000 
1.00001 


1.00001 


1.00002 
1.00002 
1.00003 
1.00004 


1 . 00005 


1.00007 
1.00010 
1.00013 
1.00016 


1.00020 


1.00031 
1 . 00046 
1 . 00062 
1 . 00080 
00102 


0 


0.00100 
0,00200 
0 . 00300 
0.00400 


0 . 00500 


0 . 00600 
0 . 00700 
0 . 00800 
0 . 00900 


0.01000 


0.01200 
0.01400 
0.01600 
0.01800 


0.02000 


0.02500 
0.03000 
0.03500 
0.04001 
0.04502 


1.00125  0.05003 


6  1 


00180  0.06004 
1.00245  0.07006 
1.00321,0.08008 
1.00405  0.09011 


1.00500  0.10016 


1.00721  0.12028 
1 . 00982  0 . 1 4046 
1.01283  0.16069 
1.01624  0.18097 


1.02006  0.20134 


0 


0.001 
0.002 
0.003 
0.004 


0.005 


0.006 
0.007 
0.008 
0.009 


0.010 


0.012 
0.014 
0.016 
0.018 


0.020 


0.025 
0.030 
0.035 
0.040 
0.045 


0.050 


0.06 
0.07 
0.08 
0.09 


0.10 


0.12 
0.14 
0.16 
0.18 


0.20 


£  +  •""»  =  1.001000494, 


£-•"*"  =0.99900049. 


314  ENGINEERING  MATHEMATICS. 

Table  II — Continued. 

EXPONENTIAL   AND    HYPERBOLIC   FUNCTIONS. 


T 

log  e  +  i 

log  £--^ 

£  +  ^ 

£-' 

cosh  X 

sinh  X 

X 

0.20 

0.086859 

9.913141 

1.22140 

0.81873 

1.02006 

0.20134 

0.20 

0.25 
0.30 
0.35 
0.40 
0.45 

0 . 108574 
0 . 130288 
0.152003 
0.173718 
0.195433 

9.891426 
9.869712 
9.847997 
9.826282 
9.804567 

1 . 28403 
1.34986 
1.41907 
1.49183 
1.56831 

0.77880 
0.74082 
0.70469 
0.67032 
0.63763 

1.03142 
1.04534 
1.06188 
1.08108 
1 . 10297 

0.25261 
0.30457 
0.35719 
0.41076 
0.46534 

0.25 
0.30 
0.35 
0.40 
0.45 

0.50 

0.217147 

9.782853 

1.64870 

0.60653 

1.12761 

0.52108 

0.50 

0.6 
0.7 
0.8 
0.9 

0.260577 
0.304006 
0.347436 
0 . 390865 

9.739423 
9.695994 
9 . 652564 
9.609135 

1.82212 
2.01375 
2 . 22554 
2 .  4596() 

0.54881 
0.49659 
0 . 44933 
0.40657 

1.19546 
1.25517 
1.33744 
1.43309 

0.63666 
0.75858 
0.88811 
1.02657 

0.6 
0.7 
0.8 
0.9 

1.0 

0.434294 

9.565706 

2.71828 

0.36788 

1.54308 

1 . 17520 

1.0 

1.2 
1.4 
1.6 
1.8 

0.521153 
0.608012 
0.694871 
0.781730 

9.478847 
9.391988 
9.305129 
9.218270 

3.32011 
4 . 0552C 
4 .  9530-1 
6 . 04965 

0.30119 
0.24660 
0.20190 
0.16530 

1.81065 
2.15090 
2.57745 
3.10745 

1.50946 
1.90430 
2.37557 
3.44218 

1.2 
1,4 
1.6 
1.8 

2.0 

0.868589 

9.131411 

7 . 38906 

0.13534 

3.76220 

3.62686 

2.0 

2.5 
3.0 
3.5 

4.0 
4.5 

1.085736 
1.302883 
1.520030 
1.737178 
1.954325 

8.914264 
8.694117 
8.479970 
8.262822 
8.045675 

12.1825 
20.0855 
33.1154 
54.5983 
90.0170 

0.082085 
0.049797 
0.030197 
0.018316 
0.011109 

6.1323 
10.0677 
16.5718 
27.3083 
45.0141 

6.0002 
10.0178 
16.5426 
27 . 2900 
■1O.0030 

2.5 
3.0 
3.5 
4.0 
4.5 

5.0 

2.171472 

7.828528 

148.413 

0 . 006738 

74.210 

74 . 203 

5.0 

6 
7 
8 
9 

2.605767 
3.040061 
3.474356 
3.908650 

7.394233 
6.959939 
6.525644 
6.091350 

103.428 
1096.63 
2980.96 
8103.08 

0.002479 
0.000912 
0.000335 
0.000123 

201.715 

201.713 

6 

7 
8 
9 

=  *£ 
for  X 

>6 

10 

4 . 342945 

5.657055 

22026.5 

0.0000454 

10 

12 
14 
16 

18 

5.211534 
6.080123 
6.948712 
7.817301 

4 . 788466 
3.919877 
3.051288 
2.182699 

162755 

120261(1 

8886 12(. 

65660000 

0.0000061 
0 . 00000083 
0.00000011 
0 . 00000002 

12 
14 
16 
18 

20 

! 

8.685890 

1.314110 

485166000 

0.00000000 

20 

INDEX 


B 


Abelian  integrals  and  functions,  305 
Absolute  number,  4 

value  of  fractional  expression,  49 
of  general  number,  30 
Accuracy,  loss  of,  281 

of  approximation  estimated,  200 
of  calculation,  279 
of  curve  equation,  210 
of  transmission  line  equations,  208 
Addition,  1 

of  general  number,  28 
and  subtraction  of  trigonometric 
functions,  102 
Algebra  of  general  number  or  com- 
plex quantity,  25 
Algebraic  expression,  294 

function,  75 
Alternating  current  and  voltage  vec- 
tor, 41 
functions,  117,  125 
waves,  117,  125 
Alternations,  117 

Alternator  short  circuit  current,  ap- 
proximated, 195 
Analytical   calculation    of   extrema, 
152 
function,  294 
Angle,  see  also  Phase  angle. 
Approximation  calculation,  280 

by  chain  fraction,  208c 
Approximations  giving  (1  +  s)  and 
(1  -  s),  201 
of  infinite  series,  53 
methods  of,  187 
Arbitrary  constants  of  series,  69,  79 
Area  of  triangle,  106 
Arrangement  of  numerical  calcula- 
tions, 275 
Attack,  method  of,  275 


Base  of  logarithm,  21 

Binomial  series  with  small  quanti- 
ties, 193 
theorem,  infinite  series,  59 
of  trigonometric  function,  104 

Biquadratic  parabola,  219 


Calculation,  accuracy,  279 
checking  of,  291 
numerical,  258 
reliability,  271 

Capacity,  65 

Chain  fraction,  208 

Change  of  curve  law,  211,  234 

Characteristics  of  exponential  curves, 
228 
of  parabolic  and  hyperbolic  curves, 
223 

Charging  current  maximum  of  con- 
denser, 176 

Checking  calculations,  293a 

Ciphers,  number  of,  in  calculations, 
282 

Circle  defining  trigonometric  func- 
tions, 94 

Coefficients,    unknown,    of    infinite 
series,  60 

Combination    of   exponential    func- 
tions, 231 
of  general  numbers,  28 
of  vectors,  29 

Comparison  of  exponential  and  hy- 
perbolic curves,  229 

Complementary  angles   in   trigono- 
metric functions,  99 

Complex  imaginary  quantities,  see 
General  number. 


315 


316 


INDEX 


Complex,  quantity,  17 
algebra,  27 
see  General  number. 
Conjugate  numbers,  31 
Constant,  arbitrary  of  series,  69,  79 
errors,  186 

factor  with  parabolic  and  hyper- 
bolic curves,  223 
phenomena,  106 
terms  of  curve  equation,  211 
of  empirical  curves,  234 
in  exponential  curves,  230 
with  exponential  curves,  229 
in     parabolic     and     hyperbolic 
curves,  225 
Convergency      determinations     of 
potential  series,  215 
of  series,  57 
Convergent  series,  56 
Coreless  by  potential  series,  213 

curve  evaluation,  244 
Cosecant  function,  98 
Cosh  function,  305 
Cosine-amplitude,  299 

components  of  wave,  121,  125 
function,  94 
series,  82 

versed  function,  98 
Cotangent  function,  94 
Counting,  1 

Current    change    curve    evaluation, 
241 
of  distorted  voltage  wave,  169 
input    of    induction    motor,    ap- 
proximated, 191 
maximum    of    alternating    trans- 
mission circuit,  159 
Curves,  checking  calculations,  2936 
empirical,  209 
law,  change,  234 
rational  equation,  210 
use  of,  284 


D 


Data  on  calculations  and  curves,  271 

derived  from  curve,  285 
Decimal  error,  2936 


Decimals,  number  of,  in  calculations, 
282 

in  logarithmic  tables,  281 
Definite   integrals   of  trigonometric 

functions,  103 
Degrees  of  accuracy,  279 
Delta-amplitude,  299 
Differential  equations,  64 

of  electrical  engineering,  65,  78,  86 

of  second  order,  78 
Differentiation      of      trigonometric 

functions,  103 
Diophantic  equations,  186 
Distorted  electric  waves,  108 
Distortion  of  wave,  139 
Divergent  series,  56 
Division,  6 

of  general  number,  42 

with  small  quantities,  190 
Double     angles     in     trigonometric 
functions,  103 

peaked  wave,  255,  260,  266 

periodicity    of    elliptic    functions, 
299 

scale,  289 


E 


21 


Efficiency  maximum  of  alternator, 
162 
of  impulse  turbine,  154 
of  induction  generator,  177 
of  transformer,  155,  174 
Electrical    engineering,    differential 

equations,  65,  78,  86 
Ellipse,  length  of  arc,  301 
Elliptic  integrals  and  functions,  299 
Empirical  curves,  209 
evaluation,  233 
equation  of  curve,  210 
Engineering    differential    equations, 
65,  78,  86 
reports,  290 
Equilateral  hyperbola,  217 
Errors,  constant,  186 
numerical,  2936 
of  observation,  180 


INDEX 


317 


Estimate  of  accuracy  of  approxima- 
tion, 200 
Evaluation  of  empirical  curves,  233 
Even  functions,  81,  98,  305 
periodic,  122 
harmonics,  117,  266 

separation,  120,  125,  134 
Evolution,  9 

of  general  number,  44 
of  series,  70 
Exact  calculation,  281 
Exciting    current    of    transformer, 

resolution,  137 
Explicit  analytic  function,  294 
Exponent,  9 
Exponential  curves,  227 

forms  of  general  number,  50 
functions,  52,  297,  304 

with  small  quantities,  196 
series,  71 

tables,  312,  313,  314 
and  trigonometric  functions,  rela- 
tion, 83 
Extrapolation  on  curve,  limitation, 

210 
Extrema,  147 

analytic  determination,  152 
graphical  construction  of  differen- 
tial function,  170 
graphical  determination,  147,  150, 

168 
with  intermediate  variables,  155 
with  several  variables,  163 
simplification  of  function,  157 


Factor,     constant,    with    parabolic 

and  hyperbolic  curves,  223 
Fan  motor  torque  by  potential  ser- 
ies, 215 
Fifth  harmonic,  261,  264 
Flat  top  wave,  255,  260,  265,  268 
zero  waves,  255,  258,  261,  265 
Fourier     series,    see     Trigonometric 

series. 
Fraction,  8 
as  series,  52 
chain-,  208 


Fractional  exponents,  11,  44 

expressions  of  general  number,  49 
Full  scale,  289 
Functions,  theory  of,  294 

G 

Gamma  function,  304 
General  number,  1,  16 
algebra,  25 

engineering  reports,  291 
exponential  forms,  50 
reduction,  48 

reports  on  engineering  matters, 
292 
Geometric    scale  of  curve    plotting, 

288 
Graphical  determination  of  extrema, 
147,  150,  168 

H 

Half  angles  in  trigonometric  func- 
tions, 103 
Half  waves,  117 
Half  scale,  289 
Harmonics,  even,  117 
odd,  117 

of  trigonometric  series,  114 
two,  in  wave,  255 
High  harmonics  in  wave  shape,  255, 

269 
Hunting  of  synchronous  machines, 

257 
Hyperbola,  arc  of,  61 

equilateral,  217 
Hyperbolic  curves,  216 
functions,  294 

curve,  shape,  232 
integrals  and  functions,  298 
tables,  313,  314 
Hyperelliptic    integrals    and    func- 
tions, 301 
Hysteresis  curve  of  silicon  steel,  in- 
vestigation of,  248 


Imaginary  number,  26 

quantity,  see  Quadrature  number. 


318 


INDEX 


Incommensurable  waves,  257 
Indeterminate  coefficients,  method, 

71 
Indeterminate    coefficients    of    infi- 
nite series,  60 
Individuals,  8 
Inductance,  65 
Infinite  series,  52 

values  of  curves,  211 
of  empirical  curves,  233 
Inflection  points  of  curves,  153 
Impedance  vector,  41 
Implicit  analytic  function,  294 
Integral  function,  295 
Integration  constant  of  series,  69,  79 

of  differential  equation,  65 

by  infinite  series,  60 

of  trigonometric  functions,  103 
Intelligibility  of  calculations,  283 
Intercepts,  defining  tangent  and  co- 
tangent functions,  94 
Involution,  9 

of  general  numbers,  44 
Irrational  numbers,  11 
Irrationality    of    representation    by 
potential  series,  213 


J,  14 


Least  squares,  method  of,  179,  186 
Limitation  of  mathematical   repre- 
sentation, 40 

of  method  of  least  squares,  186 

of  potential  series,  216 
Limiting  value  of  infinite  series,  54 
Linear  number,  33 

see  Positive  and  Negative  number. 
Line  calculation,  276 

equations,  approximated,  204 
Logarithm  of  exponential  curve,  229 

as  infinite  series,  63 

of  parabolic  and  hyperbolic  curves, 
225 

with  small  quantities,  197 


Logarithmation,  20 

of  general  numbers,  51 
Logarithmic  curves,  227 

functions,  297 

paper,  233,  287 

scale,  288 

tables,  number  of  decimals  in,  281 
Loss  of  curve  induction  motor,   183 


M 


Magnetic    characteristic    on    semi- 
logarithmic  paper,  288 
Magnetite  arc,  volt-ampere  charac- 
teristic, 239 

characteristic,  evaluation,  246 
Magnitude  of  effect,  determination, 

280 
Maximum,  see  Extremum. 
Maxima,  147 

McLaurin's  series  with  small  quan- 
tities, 198 
Mechanism  of  calculating  empirical 

curves,  237 
Methods  of  calculation,  275 

of  intermediate  coefficients,  71 

of  least  squares,  179,  186 

of  attack,  275 
Minima,  147 

Minimum,  see  Extremum. 
Multiple  frequencies  of  waves,  274 
Multiplicand,  39 
Multiplication,  6 

of  general  numbers,  39 

with  small  quantities,  188 

of  trigonometric  functions,  102 
Multiplier,  39 


N 


Negative    angles    in    trigonometric 
functions,  98 

exponents,  11 

number,  4 
Nodes  in  wave  shape,  256,  270 
Non-periodic  curves,  212 
Nozzle  efficiency,  maximum,  150 
Number,  general,  I 


INDEX 


319 


Numerical  calculations,  275 
values  of  trigonometric  functions, 
101 


O 


Observation,  errors,  180 
Octave  as  logarithmic  scale,  288 
Odd  funcfons,  81,  98,  305 

period  c,  122 
harmonics  in  symmetrical  wave, 
117 

separation,  120,  125,  134 
Omissions  in  calculations,  2936 
Operator,  40 

Order  of  small  quantity,  188 
Oscillating  functions,  92 
Output,  see  Power. 


v  and  7y  added  and  subtracted  in 

trigonometric  function,   100 
approximated  by  chain  fraction, 
208c 
Pairs  of  high  harmonics,  270 
Parabola,  common,  218 
Parabolic  curves,  216 
Parallelogram  law  of  general  num- 
bers, 28 
of  vectors,  29 
Peaked  wave,  255,  258,  261,  264 
Pendulum  motion,  301 
Percentage  change  of  parabolic  and 

hyperbolic  curves,  223 
Periodic  curves,  254 
decimal  fraction,  12 
phenomena,  106 
Periodicity,  double,  of  elliptic  func- 
tions, 299 
of  trigonometric  functions,  96 
Permeability  maximum,  148,  170 
Phase  angle  of  fractional  expression, 
49 
of  general  number,  28 
Plain  number,  32 

see  General  nuinber. 


Plotting  of  curves,  212 

proper  and  improper,  286 
of  empirical  curve,  234 
Polar  co-ordinates  of  general  num- 
ber, 25,  27 
expression  of  general  number,  25, 
27,  38,  43,  44,  48 
Polyphase  relation,  reducing  trigo- 
nometric series,  134 
of  trigonometric  functions,  104 
system  of  points  or  vectors,  46 
Positive  number,  4 
Potential  series,  52,  212 
Power  factor  maximum  of  induction 
motor,  149 
maximum    of    alternating    trans- 
mission circuit,  158 
of  generator,  161 
of  shunted  resistance,  155 
of  storage  battery,  172 
of  transformer,  173 
of  transmission  line,  165 
not  vector  product,  42 
of  shunt  motor,  approximated,  189 
with  small  quantities,  194 
Probability  calculation,  181 
Product  series,  303 

of  trigonometric  functions,  102 
Projection,  defining  cosine  function, 

94 
Projector,  defining  sine  function,  94 


Q 


Quadrants,    sign    of    trigonometric 

functions,  96 
Quadrature  numbers,  13 
Quarter  scale,  289 
Quaternions,  22 


R 


Radius  vector  of  general  number,  28 
Range  of  convergency  of  series,  56 
Ratio  of  variation,  226 
Rational  equation  of  curve,  210 

function,  295 
Rationality  of  potential  series,  214 


320 


INDEX 


Real  number,  26 

Rectangular  co-ordinates  of  general 

number,  25 
Reduction  to  absolute  values,  48 
Relations     of     hyperbolic     trigono- 
metric and  exponential  func- 
tions, 309 
Relativeness  of  small  quantities,  188 
Reliability  of  numerical  calculations, 

293 
Reports,  engineering,  290 
Resistance,  65 
Resolution  of  vectors,  29 
Reversal  by  negative  unit,  14 

double,  at  zero  of  wave,  258,  261 
Reverse  function,  294 
Right  triangle  defining  trigonomet- 
ric functions,  94 
Ripples  in  wave,  45 

by  high  harmonics,  270 
Roots  of  general  numbers,  45 

expressed  by  periodic  chain  frac- 
tion, 208e 
with  small  quantities,  194 
of  unit,  18,  19,  46 
Rotation  by  negative  unit,  14 
by  quadrature  unit,  14 


S 


Saddle  point,  165 

Saw-tooth  wave,  246,  255,  258,  260, 

265 
Scalar,  26,  28,  30 

Scale  in  curve  plotting,  proper  and 
improper,  212,  286 

full,  double,  half,  etc.,  287 
Scientific  engineering  records,  291 
Secant  function,  98 
Second  harmonic,  effect  of,  266 
Secondary  effects,  210 

phenomena,  234 
Semi-logarithmic  paper,  287 
Series,  exponential,  71 

infinite,  52 

trigonometric,  106 
Seventh  harmonic,  262 


Shape  of  curves,  212 

proper  in  plotting,  286 

of  exponential  curve,  227,  230 

of  function,  by  curve,  284 

of  hyperbolic  functions,  232 

of  parabolic  and  hyperbolic  curves, 
217 
Sharp  zero  wave,  255,  260,  265 
Short  circuit  current  of  alternator, 

approximated,  195 
Sign  error,  293c 

of  trigonometric  functions,  95 
Silicon  steel,  investigation  of  hystere- 
sis curve,  248 
Simplification  by  approximation,  187 
Sine-amplitude,  199 

component  of  wave,  121,  125 

function,  94 

series,  82 

versus  function,  98 
Sine  function,  305 
Slide  rule  accuracy,  281 
Small  quantities,  approximation,  187 
Special  functions,  302 
Squares,  least,  method  of,  179,  186 
Steam  path  of  turbine,  33 
Subtraction,  1 

of  general  number,  28 

of  trigonometric  functions,  102 
Summation  series,  303 
Superposition  of  high  harmonics,  273 
Supplementary    angles    in    trigono- 
metric functions,  99 
Surging    of   synchronous   machines, 

301 
Symmetrical  curve  maximum,  150 

periodic  function,  117 

wave,  117 


Tabular  form  of  calculation,  275 

Tangent  function,  94 

Taylor's  series  with  small  quantities, 

199 
Temperature  wave,  131 
Temporary  use  of  potential  series, 

216 


INDEX 


321 


Terminal  conditions  of  problem,  69 
Terms,  constant,  of  empirical  curves, 
234 
in  exponential  curve,  229 
with  exponential  curve,  229 
in     parabolic     and    hyperbolic 
curves,  225 
of  infinite  series,  53 
Theorem,    binomial,    infinite  series, 

59 
Thermomotive  force  wave,  133 
Theta  functions,  300 
Third  harmonic,  136,  255 
Top,  peaked  or  flat,  of  wave,  255 
Torque  of  fan  motor  by  potential 

series,  215 
Transient  current  curve,  evaluation, 
241 
phenomena,  106 
Transmission    equations,     approxi- 
mated, 204 
line  calculation,  275 
Treble  peak  of  wave,  262 
Triangle,      defining      trigonometric 
functions,  94 
trigonometric  relations,  106 
Trigonometrical     and     exponential 
functions,  relations,  83 
functions,  94,  304 
series,  82 

with  small  quantity,  198 
integrals  and  functions,  298 
series,  106 

calculation,  114,  116,  139 
Triple  harmonic,  separation,  136 
peaked  wave,  255 
scale,  289 


Tungsten      filament,      volt-ampere 

characteristic,  235 
Turbine,  steam  path,  33 


u 


Unequal  height  and  length  of  half 

waves,  268 
Univalent  functions,  106 
Unsymmetric  curve  maximum,  151 
wave,  138 


Values  of  trigonometric   functions, 

101 
Variation,  ratio  of,  226 
Vector  analysis,  32 
multiplication,  39 
quantity,  32 

see  General  number. 
representation  by  general  number, 
29 
Velocity  diagram  of  turbine  steam 
path,  34 
functions  of  electric  field,  304 
Versed    sine   and   cosine   functions, 

98 
Volt-ampere  characteristic  of  mag- 
netite arc,  239 
of  tungsten  filament,  235 


Zero  values  of  curve,  211 
of  empirical  curves,  233 
of  waves,  255 


SCIENCE  AND  ENGINEERING 

LIBRARY 

University  of  California, 

San   Diego 


SEpirm'l''' 


JAN  1  5 1986 


FEB  0  5  1986 


'Mm 


SE  16 


UCSD  Libr. 


UC  SOUTHERN  REGIONAL  LIBRARY  FACILITY 


AA    001  294  508  5 


